The Variety of Integrable Killing Tensors on the 3-Sphere

Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere $S^3$ and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all St\"ackel systems in these varieties. This allows us to recover the known list of separation coordinates on $S^3$ in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron $K_4$.

1. Introduction 1.1. The problem. Given a partial differential equation, it is a natural problem to seek for coordinate systems in which this equation can be solved by a separation of variables and to classify all such coordinate systems. It is not surprising, that for classical equations this problem has a very long history and is marked by the contributions of a number of prominent mathematicians, beginning with Gabriel Lamé in the first half of of the 19th century [Lam37]. For example, consider the Hamilton-Jacobi equation on a Riemannian manifold. We say that it separates (additively) in a system of coordinates x 1 , . . . , x n if it admits a solution of the form depending on n constants c = (c 1 , . . . , c n ). If this is the case, the initial partial differential equation decouples into n ordinary differential equations. We call the corresponding coordinate system separation coordinates. It turns out that the Hamilton-Jacobi equation in the form (1.1) separates in a given orthogonal coordinate system if and only if it also separates for V ≡ 0 and if the potential V satisfies a certain compatibility condition. In fact, the Hamilton-Jacobi equation with V ≡ 0 plays a key role for separation of variables in general. The reason is that for orthogonal coordinates the (additive) separation of this equation is a necessary condition for the (multiplicative) separation of other classical equations such as the Laplace equation, the Helmholtz equation or the Schrödinger equation. Sufficient conditions can be given in the form of compatibility conditions for the curvature tensor and for the potential.
The classification of orthogonal separation coordinates for the Hamilton-Jacobi equation thus leads to a variety of ordinary differential equations. These are wellknown equations that define special functions appearing all over in mathematics and physics -such as Bessel functions, Legendre polynomials, spherical harmonics or Mathieu functions, to name just a few. In particular, these functions serve as bases for series expansions in the explicit solution of boundary value problems on domains bounded by coordinate hypersurfaces.
Separation coordinates are intimately related to the existence of rank two Killing tensors that satisfy a certain integrability condition.
Definition 1.1. A symmetric tensor K αβ on a Riemannian manifold is a Killing tensor if and only if where ∇ is the Levi-Civita connection of the metric g and the round parenthesis denote complete symmetrisation of the enclosed indices. Depending on the context, we will consider a Killing tensor either as a symmetric bilinear form K αβ or alternatively as a symmetric endomorphism K α β = g αγ K γβ .
We say a symmetric endomorphism K is integrable, if around any point in some open and dense set we find local coordinates such that the coordinate vectors are eigenvectors of K. This geometric definition can be cast into an analytic form with the aid of the Nijenhuis torsion of K, defined by (1.3) Integrability in the above sense is then equivalent to the following system of nonlinear first order partial differential equations [Nij51]. We take this characterisation as the definition of integrability.
Definition 1.2. A symmetric endomorphism field K α β on a Riemannian manifold is integrable if and only if it satisfies the Nijenhuis integrability conditions where the square brackets denote complete antisymmetrisation in the enclosed indices.
At the end of the 19th century, Paul Stäckel showed that any system of orthogonal separation coordinates gives rise to n − 1 linearly independent, mutually commuting integrable Killing tensors with simple eigenvalues [Stä97]. And in 1934, Luther P. Eisenhart proved the converse [Eis34]. Notice that the metric is a trivial example of an integrable Killing tensor and it commutes with any other Killing tensor, but does not have simple eigenvalues. Definition 1.3. A Stäckel system on an n-dimensional Riemannian manifold is an n-dimensional space of mutually commuting integrable Killing tensors.
It can be shown that any integrable Killing tensor with simple eigenvalues is contained in a Stäckel system and vice versa, any Stäckel system contains such a Killing tensor [Ben97]. This establishes a one-to-one correspondence between orthogonal separation coordinates and Stäckel systems.
The Nijenhuis integrability conditions (1.4) are homogeneous algebraic equations in K and ∇K, which are invariant under isometries. Separation of variables therefore naturally leads to the following problem.
Problem I. Determine the projective variety K(M ) of all integrable Killing tensors on a given Riemannian manifold M . That is, solve explicitly the Nijenhuis integrability conditions (1.4) for Killing tensors (1.2). Moreover, find natural isometry invariants characterising the integrability of a Killing tensor.
By the definition of integrability, any Killing tensor in dimension one and two is integrable. Therefore, Problem I is trivial in dimensions less than three, once the space of Killing tensors is known. However, it has never been solved in higher dimensions, not even for the least non-trivial examples, namely Euclidean space R 3 , the sphere S 3 or hyperbolic space H 3 . Moreover, an explicit solution of the equations (1.4) has been considered intractable by various experts in this domain [HMS05].
Although a general solution of Problem I is unknown for any non-trivial example, there is a construction which yields a large family of Stäckel systems. This family is interesting in itself, because it appears under different names in a number of different guises in the theory of integrable systems: as "Newtonian systems of quasi-Lagrangian type" [RWML99], as "systems admitting special conformal Killing tensors" [Cra03a], as "cofactor systems" [Lun01], as "bi-quasi-Hamiltonian systems" [CS02,Cra03b], as "L-systems" [Ben05] or as "Benenti systems" [BM03]. We follow the latter nomenclature of Bolsinov and Matveev, because their description is best suited to our context. This brings us to the following question.
Question I. Do Benenti systems parametrise the entire variety of integrable Killing tensors?
Moreover, by construction, the set of Benenti systems is not only invariant under isometries, but even under the projective group. So, if the answer to Question I is "no" and the isometry group is a proper subgroup of the projective group, the following question arises.
In this article we will completely solve Problems I to IV for the three dimensional sphere S 3 equipped with the round metric and give answers to the questions I and II. This is done without relying on any computer algebra. One of the reasons we choose S 3 is that it has the simplest isometry group among the 3-dimensional constant curvature manifolds. Our solution will reveal diverse facets of the problem of separation of variables, related to representation theory, algebraic geometry, geometric invariant theory, geodesic equivalence, Stasheff polytopes or moduli spaces of stable algebraic curves.
1.2. Prior results. In his fundamental paper Eisenhart also derived a list of orthogonal separation coordinates for R 3 and S 3 [Eis34]. These results have been generalised by Kalnins & Miller, who gave an iterative diagrammatical procedure for the construction of orthogonal separation coordinates on spaces of constant curvature in any dimension [Kal86,KM86]. Furthermore, Eisenhart computed the Stäckel systems on R 3 in their respective separation coordinates. Despite these advances, the topology or the geometry of the space of integrable Killing tensors or its quotient by isometries has never been understood for any non-trivial example.
Since explicit solutions were out of sight, the use of computer algebra increased in recent years. This has lead to the construction of a set of 15 isometry invariants for Killing tensors on R 3 which discriminate between the different types of orthogonal separation coordinates [HMS05,Hor08]. The drawback of this method is that it depends on an a priori knowledge of separation coordinates and can therefore not be generalised. Besides, it is highly computational and thus unsuitable for higher dimensions. Nothing similar is known for the sphere or hyperbolic space.

Method & results.
Our method is based on a purely algebraic description of the vector space of Killing tensors on constant curvature manifolds in combination with the corresponding algebraic equivalent of the Nijenhuis integrability conditions. We consider the standard 3-sphere, isometrically embedded as the unit sphere in a 4-dimensional Euclidean vector space (V, g): The vector space of Killing tensors K on S 3 is then naturally isomorphic to the vector space of algebraic curvature tensors on the ambient space V [MMS04]. These are (constant) tensors R a1b1a2b2 on V having the symmetries of a Riemannian curvature tensor. The isomorphism is explicitly given by x ∈ S 3 , v, w ∈ T x S 3 , (1.6) and equivariant with respect to the natural actions of the isometry group O(V, g) on both spaces. Under this correspondence the Nijenhuis integrability conditions (1.4) for a Killing tensor are equivalent to the following algebraic integrability conditions for the associated algebraic curvature tensor [Sch12]: Here the Young symmetrisers on the left hand side denote complete antisymmetrisation in the indices a 2 , b 2 , c 2 , d 2 respectively complete symmetrisation in the indices a 1 , b 1 , c 1 , d 1 . These homogeneous algebraic equations define the space of integrable Killing tensors as a projective variety.
The key to the solution of the algebraic integrability conditions is a reinterpretion in terms of the decomposition of algebraic curvature tensors into irreducible representations of the isometry group -the selfdual and anti-selfdual Weyl part, the trace-free Ricci part and the scalar part. Under the action of the isometry group these components can be brought to a certain normal form, which simplifies considerably if the algebraic integrability conditions are imposed.
We regard an algebraic curvature tensor on V as a symmetric endomorphism on Λ 2 V . In this sense we will say that an algebraic curvature tensor R is diagonal if R ijkl = 0 unless {i, j} = {k, l}. In dimension four a diagonal algebraic curvature tensor can be parametrised as where (α, β, γ) denotes any cyclic permutation of (1, 2, 3). In this parametrisation the w α are the eigenvalues of the self-dual Weyl part, which turn out to be the same as those of the anti-self-dual Weyl part, the t α parametrise the eigenvalues of the trace free Ricci tensor and s is the scalar curvature. In a first step we prove that diagonal algebraic curvature tensors define a (local) slice for the action of the isometry group on integrable Killing tensors.
Theorem 1.4 (Integrability implies diagonalisability). Under the action of the isometry group any integrable Killing tensor on S 3 is equivalent to one with a diagonal algebraic curvature tensor.
The first algebraic integrability condition is identically satisfied on diagonal algebraic curvature tensors and the second reduces to a single algebraic equation on the diagonal entries. This yields a description of integrable Killing tensors with diagonal algebraic curvature tensor as a projective variety, equipped with the residual action of the isometry group. We call this variety the Killing-Stäckel variety (KS-variety).
Theorem 1.5 (Integrability in the diagonal case). A Killing tensor on S 3 with diagonal algebraic curvature tensor (1.8) is integrable if and only if the associated matrix    has zero trace and zero determinant, where This defines a projective variety in P 4 which carries a natural S 4 -action, given by conjugating (1.9) with the symmetries of the regular octahedron in R 3 with vertices ±e i and adjacent faces oppositely oriented. Two diagonal algebraic curvature tensors are equivalent under the isometry group if and only if their associated matrices (1.9) are equivalent under this S 4 -action.
Combining Theorems 1.4 and 1.5 yields a solution to Problem II: The projectivisation of the quotient (1.5) is isomorphic to the quotient of the KS-variety by its natural S 4 -action. Moreover, in the course of the proof of both theorems we establish isometry invariants for the integrability of a Killing tensor.
Proposition 1.6 (Isometry invariants). Let W and T be the Weyl and Ricci part of the algebraic curvature tensor corresponding to a Killing tensor on S 3 ⊂ V , regarded as endomorphisms on Λ 2 V . Then this Killing tensor is integrable if and only if (i) tr [W, T] 2 = 0 (ii) tr ( * W 2 ) = tr ( * W 3 ) = 0, where * denotes the Hodge operator on Λ 2 V and (iii) the endomorphisms I, W and W 2 − T 2 are linearly dependent.
With this result the check whether a given Killing tensor on S 3 is integrable becomes algorithmically trivial.
Naïvely, one can find the Stäckel systems by solving the commutator equation [K,K] = 0 for a fixed integrable Killing tensor K. But a far simpler way is to exploit the algebraic geometry of the KS-variety. Recall that Stäckel systems are linear spaces of dimension n and always contain the metric. We show that the Killing tensors in a Stäckel system have simultaneously diagonalisable algebraic curvature tensors. Therefore Stäckel systems on S 3 correspond to certain projective lines in the KS-variety. In general, the projective lines in a projective variety constitute themselves a projective variety, called the Fano variety. That is, we are looking for a subvariety of the Fano variety of the KS-variety. By definition, the KS-variety is a linear section of the variety of 3 × 3 matrices with vanishing determinant, the so called full determinantal variety. Fortunately, the Fano variety of a full determinantal variety is well understood just in the case we need here.
So we only have to use the commutator equation to check which projective lines correspond to Stäckel systems, rather than to actually solve it. In order to do so, we derive an algebraic equivalent of the commutator equation in terms of the corresponding algebraic curvature tensors. This entails a detailed description of the algebraic geometry of the KS-variety and the Stäckel systems therein. In particular, we can relate the algebraic geometric properties of the KS-variety to the geometric properties of the corresponding integrable Killing tensors. For this, we remark that there is a natural way how to extend (integrable) Killing tensors from S m to S n for m < n.
Theorem 1.7. For M = S 3 the projectivisation of the quotient (1.5) is isomorphic to the quotient of the KS-variety by its natural action of S 4 . Furthermore: (i) The singularities of the KS-variety are extensions of the metric on S 2 respectively on S 1 . (ii) Stäckel systems determine projective lines in the KS-variety which consist of matrices (1.9) annihilating a given vector. (iii) Projective planes in the KS-variety that consist of matrices (1.9) annihilating a fixed vector correspond to extensions of (integrable) Killing tensors from S 2 to S 3 . (iv) The projective plane of antisymmetric matrices in the KS-variety corresponds to the projective space of special Killing tensors. These are Killing tensors K of the form where L is a special conformal Killing tensor 1 .
Based on the algebraic description of Stäckel systems we derive an algebraic description of Benenti systems as well and answer Question I. The answer is "almost", in the following sense. On one hand, there are Stäckel systems on S 3 , which are not Benenti systems. So the answer is clearly "no". But on the other hand these are exactly the extensions of Killing tensors from S 2 and for S 2 the answer is easily seen to be "yes".
From the algebraic description of Benenti systems we derive the following parametrisation of K(S 3 ), which is equivariant under the isometry group. This solves Problem I. Theorem 1.8 (Equivariant parametrisation). The algebraic variety K(S 3 ) is the closure of the set of algebraic curvature tensors of the form Here denotes the Kulkarni-Nomizu product 2 and 1 2 g g is the algebraic curvature tensor of the metric. That is, K(S 3 ) is the closure of the projective cone from the metric over the image of the Kulkarni-Nomizu square h → h h, a map from symmetric tensors to algebraic curvature tensors. Remark that the apex lies itself in this image.
Since the set of algebraic curvature tensors of the form h h is invariant under the projective group P GL(V ) of S 3 , the answer to Question II is "almost" as well. The reason is that although K(S 3 ) is not projectively invariant, this "invariant part" still contains the essential part of each Stäckel system unless it is an extension from S 2 . But again, on S 2 the answer is easily seen to be "yes". Our answers to Questions I and II indicate that one should think of the space of integrable Killing tensors on the sphere as composed of strata, where the non-generic strata come from lower dimensional spheres. Then each stratum is generated by Benenti systems and invariant under the projective group in the respective dimension.
As a simple consequence of Theorem 1.7 we get a canonical representative in each Stäckel system. Corollary 1.9. Each Stäckel system on S 3 contains an essentially unique special Killing tensor.
If the eigenvalues of the corresponding special conformal Killing tensor are simple, they define the respective separation coordinates [Cra03a]. Moreover, every special conformal Killing tensor L on S n is the restriction of a constant symmetric tensorL on the ambient space V [MM10]. This is the reason why we can solve the eigenvalue problem for integrable Killing tensors explicitly. In this manner we get a purely algebraic classification of separation coordinates, which solves Problem IV. The details will be given later.
A solution to Problem III can be derived from the description of Stäckel systems in the KS-variety. This gives the classification space for separation coordinates on S 3 modulo isometries.
Theorem 1.10 (Moduli space of separation coordinates). The space of Stäckel systems on S 3 modulo isometries is homeomorphic to the quotient of the blow-up of P 2 in the four points (±1 : ±1 : ±1) ∈ P 2 under the natural action of the symmetry group S 4 of a tetrahedron in R 3 inscribed in a cube with vertices (±1, ±1, ±1).
The fundamental domain of the S 4 -action is a pentagon bounded by the nongeneric separation coordinates. In fact, the multiplicities of the eigenvalues of the constant symmetric tensorL on V identify this pentagon to the associahedron (or Stasheff polytope) K 4 [Sta63]. These observations suggests a relation between separation coordinates on spheres and moduli spaces of stable curves of genus zero with marked points [DM69,Knu83]. Indeed, our thorough analysis of the case S 3 presented here has led to the following generalisation of Theorem 1.10 [SV13].
Theorem 1.11. The Stäckel systems on S n with diagonal algebraic curvature tensor form a smooth projective variety isomorphic to the real Deligne-Mumford-Knudsen moduli spaceM 0,n+2 (R) of stable genus zero curves with n + 2 marked points.
1.4. Generalisations. Most of our results are stated for constant curvature manifolds of arbitrary dimension. What impedes a straightforward generalisation to higher dimensional spheres is that in the proof of diagonalisability we make essential use of the Hodge decomposition which only exists in dimension four. Theorem 1.11 together with a proof of the following conjecture would show, that the classification space for separation coordinates on S n is the associahedron K n+1 .
Conjecture 1.12. Under the action of the isometry group any integrable Killing tensor on S n is equivalent to one with a diagonal algebraic curvature tensor.
In Lorentzian signature the Hodge star squares to minus one. A generalisation to hyperbolic space therefore naturally requires complexification. But over the complex numbers, all signatures are equivalent. This indicates that one should solve the algebraic integrability conditions over the complex numbers. The variety of complex solutions then contains not only the solution for hyperbolic space, but for all (non-flat) pseudo-Riemannian constant curvature spaces as real subvarieties, singled out by appropriate choices of a real structure. In dimension three there is an elegant way to do this using spinors and the Petrov classification of curvature tensors [MS12]. The main complication comes from the fact that diagonalisability does not hold anymore in the complexified case.
The biggest challenge remains without doubt a non-trivial classification of integrable Killing tensors and orthogonal separation coordinates on some non-constant curvature manifold. This is completely out of scope of common, purely geometric techniques. Promising candidates for the application of our algebraic approach are spaces of constant holomorphic sectional curvature, semi-simple Lie groups, homogeneous spaces or Einstein manifolds. We propose a three-stage procedure, bearing in mind S 3 as a guiding example: (i) Identify the isomorphism class of the representation of the isometry group on Killing tensors and find an explicit isomorphism (as in [MMS04] for constant curvature manifolds). (ii) Translate the Nijenhuis integrability conditions for Killing tensors into purely algebraic integrability conditions (as in [Sch12] for constant curvature manifolds). (iii) Solve the algebraic integrability conditions by means of representation theory and algebraic geometry (as in this article for S 3 ). We consider the present work as a proof of concept that this algebraic approach to the problem of separation of variables is viable.
1.5. Structure of the article. The article is organised as follows: In Section 2 we examine some properties of algebraic curvature tensors that will be used to solve the algebraic integrability conditions in Section 3. We then describe the algebraic geometry of this solution in Section 4 and identify the Stäckel systems therein in Section 5. In Section 6 we give various geometric constructions for integrable Killing tensors and interpret them algebraically. In the last section we demonstrate how the classification of separation coordinates can be recovered within our framework.
Acknowledgements: I would like to thank Robert Milson for his motivation and numerous comments during inspiring discussions about my findings. He in particular spotted a missing equation that follows from the second algebraic integrability condition. I would also like to thank Alexander P. Veselov for pointing out the link between my solution and moduli spaces of stable curves.

Properties of algebraic curvature tensors
Algebraic curvature tensors on a vector space V are four-fold covariant tensors R a1b1a2b2 having the same algebraic symmetries as a Riemannian curvature tensor: pair symmetry: Bianchi identity: Given a scalar product g on V , we can raise and lower indices. The symmetries (2.1a) and (2.1b) then allow us to regard an algebraic curvature tensor R a1a2b1b2 on V as a symmetric endomorphism R a1b1 a2b2 on the space Λ 2 V of 2-forms on V . Since we will frequently change between both interpretations, we denote endomorphisms by the same letter in boldface.
2.1. Decomposition. In the special case where dim V = 4, the Hodge star operator " * " defines a decomposition We can therefore write an algebraic curvature tensor R and the Hodge star as block matrices with the 3×3-blocks satisfying (2.2d) Here I + and I − denote the identity on Λ 2 + V respectively Λ 2 − V . The conditions (2.2b) assure symmetry, condition (2.2c) says that s = 2 tr R and condition (2.2d) is a reformulation of the Bianchi identity (2.1c). Indeed, the Bianchi identity is equivalent to the vanishing of the antisymmetrisation of R a1b1a2b2 in all four indices. In dimension four this can be written as ε a2b2a1b1 R a1b1a2b2 = 0 or ε a2b2 a1b1 R a1b1 a2b2 = 0 , where ε a2b2a1b1 is the totally antisymmetric tensor. Notice that ε a2b2 a1b1 is nothing else than the Hodge star operator and hence tr( * R) = 0 , where tr : End(Λ 2 V ) → R is the usual trace. Now (2.2d) is (2.3) applied to R in (2.2). The space of algebraic curvature tensors is an irreducible GL(V )-representation and (2.2) gives a decomposition of this representation into irreducible representations of the subgroup SO(V, g) when dim V = 4. As the notation already suggests, we can relate these components to the familiar Ricci decomposition of an algebraic curvature tensor R into • a scalar part S a1b1a2b2 := s 12 (g a1a2 g b1b2 − g a1b2 g b1a2 ) given by the scalar curvature • a trace free Ricci part T a1b1a2b2 = 1 2 (T a1a2 g b1b2 − T a1b2 g b1a2 − T b1a2 g a1b2 + T b1b2 g a1a2 ) given by the trace free Ricci tensor T a1a2 = g b1b2 R a1b1a2b2 − s 4 g a1a2 • and a Weyl part, given by the totally trace free Weyl tensor W := R−T −S.
It is not difficult to check that [ * , T] = 0 and hence We see that W + and W − are the self-dual and anti-self-dual part of the Weyl tensor. Implicitly, the above interpretation also provides an isomorphism between trace-free Ricci tensors and homomorphisms from self-dual to anti-self-dual 2-forms.
2.2. The action of the isometry group. The Killing tensor equation (1.2) as well as the integrability conditions (1.4) are invariant under the action of the isometry group. In other words, the variety of integrable Killing tensors is invariant under isometries. This will allow us later to put the algebraic curvature tensor of an integrable Killing tensor on S 3 to a certain normal form and to solve the algebraic integrability conditions for this particular form. Recall that the isometry group of S n ⊂ V is O(V ) and that SO(V ) is the subgroup of orientation preserving isometries. We first examine the induced action of SO(V ) on algebraic curvature tensors of the form (2.2). The standard action of SO(V ) on V induces a natural action on Λ 2 V which is the adjoint action of SO(V ) on its Lie algebra under the isomorphism so(V ) ∼ = Λ 2 V . Now consider the following commutative diagram of Lie group morphisms for dim V = 4: Here the double covering π in the second row is induced from the exceptional isomorphism in the first row via the universal covering maps (vertical). Under the induced isomorphism π * of Lie algebras, , the adjoint action of an element U ∈ SO(V ) corresponds to the adjoint actions of U + ∈ SO(Λ 2 + V ) and U − ∈ SO(Λ 2 − V ). Since Λ 2 + V has dimension three, the Hodge star operator gives an isomorphism under which the adjoint action of U + corresponds to the standard action on Λ 2 + V and similarly for U − . The Hodge decomposition completes the above isomorphisms to a commutative diagram which shows that the natural action of U ∈ SO(V ) on Λ 2 V is given by Hence the natural action of U on End(Λ 2 V ) is given by conjugation with this matrix. Restricting to algebraic curvature tensors we get: Proposition 2.1 (Action of the isometry group). An element U ∈ SO(V ) acts on algebraic curvature tensors in the form (2.2) by conjugation with (2.6), i. e. via . We can describe the twofold cover π explicitly in terms of orthonormal bases on the spaces V , Λ 2 + V and Λ 2 − V . It maps an orthonormal basis (e 0 , e 1 , e 2 , e 3 ) of V to the orthonormal bases (η +1 , η +2 , η +3 ) of Λ 2 for each cyclic permutation (α, β, γ) of (1, 2, 3). From this description also follows that ker π = {±I}.
The action of O(V ) on algebraic curvature tensors is now determined by the action of some orientation reversing element in O(V ), say the one given by reversing the sign of e 0 and preserving e 1 , e 2 and e 3 . This element maps η ±α to −η ∓α . Hence its action on algebraic curvature tensors in the form (2.2) is given by conjugation with (2.9) 2.3. Aligned algebraic curvature tensors. W + and W − are symmetric and hence simultaneously diagonalisable under the action (2.7) of SO(V ). On the other hand, the singular value decomposition shows that T ± is also diagonalisable under this action, although in general not simultaneously with W + and W − . The following lemma gives a criterion when this is the case.
Lemma 2.2. Let W + and W − be symmetric endomorphisms on two arbitrary Euclidean vector spaces Λ + respectively Λ − and suppose the linear map T ± : Λ + → Λ − satisfies Then there exist orthonormal bases for Λ + and Λ − such that W + , W − and T ± are simultaneously diagonal with respect to these bases.
Proof. It suffices to show that we can chose diagonal bases for W + and W − such that the matrix of T ± has at most one non-zero element in each row and in each column, for the desired result can then be obtained by an appropriate permutation of the basis elements. The above condition implies that T ± maps eigenspaces of W + to eigenspaces of W − with the same eigenvalue. Without loss of generality we can thus assume that Λ + and Λ − are eigenspaces of W + respectively W − with the same eigenvalue. But then W + and W − are each proportional to the identity and therefore diagonal in any basis. In this case the lemma follows from the singular value decomposition of T ± .
Lemma 2.3. The following conditions are equivalent for an algebraic curvature tensor (2.2): Proof. The equivalence of (i) and (ii) follows from (2.4). For the equivalence of (ii) and (iii) it suffices to note that W and T are both symmetric and hence diagonalisable.
The following terminology is borrowed from general relativity.
Definition 2.4. We say that an algebraic curvature tensor is aligned, if it satisfies one of the equivalent conditions in Lemma 2.3.
By Lemma 2.3, for any aligned algebraic curvature tensor we can find an orthonormal basis of V such that (2.10b) To simplify notation we agree that henceforth the indices (α, β, γ) will stand for an arbitrary cyclic permutation of (1, 2, 3). Changing the basis in Λ 2 V from (2.8) to e i ∧ e j , 0 i < j 4, we obtain the independent components of an aligned algebraic curvature tensor: (2.11) The trace free Ricci tensor of this aligned algebraic curvature tensor is diagonal and given by This is nothing else than the restriction of the isomorphism (2.5) to diagonal tensors.
Definition 2.5. We call an algebraic curvature tensor on V diagonal in a basis e i if it is diagonal as an element of End(Λ 2 V ) with respect to the associated basis Of course, being a symmetric endomorphism on Λ 2 V , an algebraic curvature tensor is always diagonalisable under the full group SO(Λ 2 V ).
Proposition 2.6 (Diagonalisability criterion). An algebraic curvature tensor is diagonalisable if and only if it is aligned and W + has the same characteristic polynomial as W − .
Proof. Consider an aligned algebraic curvature tensor and suppose that W + has the same characteristic polynomial as W − . Then we can find a transformation (2.7) such that, as 3 × 3 matrices, W + = W − . The condition W − T ∓ = T ± W + then implies that T ± can be simultaneously diagonalised with W + = W − . This means we can assume without loss of generality that w +α = w −α =: w α for α = 1, 2, 3 in (2.10), so that (2.11) reads Obviously this algebraic curvature tensor is diagonal. On the other hand, any diagonal algebraic curvature tensor can be written in the form (2.13), because (2.14c) 2.5. The residual action of the isometry. For later use we need the stabiliser of the space of diagonal algebraic curvature tensors under the action of the isometry group. Such tensors are of the form (2.10) with w −α = w +α and are invariant under the transformation (2.9). Hence it is sufficient to consider only the subgroup SO(V ) of orientation preserving isometries. Let U ∈ SO(V ) with π(U ) = (U + , U − ) be an element preserving the space of diagonal algebraic curvature tensors under the action (2.7). Then In the orthogonal group, the stabiliser of the space of diagonal matrices under conjugation is the subgroup of signed permutation matrices. They act by permuting the diagonal elements in disregard of the signs. In particular, in SO(3) the stabiliser subgroup of the space of diagonal matrices on R 3 is isomorphic to S 4 . Under the isomorphism S 4 ∼ = S 3 V 4 , the permutation group S 3 is the subgroup of (unsigned) permutation matrices and acts by permuting the diagonal elements, whereas the Klein four group V 4 is the subgroup of diagonal matrices in SO(3) and acts trivially.
Using the above, it is not difficult to see that the stabiliser of the space of diagonal algebraic curvature tensors under the action (2.7) is isomorphic to S 4 × ker π, with ker π = {±I} acting trivially. Under the isomorphism S 4 ∼ = S 3 V 4 , the factor S 3 acts by simultaneous permutations of (w 1 , w 2 , w 3 ) and (t 1 , t 2 , t 3 ) and the factor V 4 by simultaneous flips of two signs in (t 1 , t 2 , t 3 ).

Solution of the algebraic integrability conditions
This section is dedicated to proving Theorems 1.4 and 1.5.
3.1. Reformulation of the first integrability condition. In the same way as for the Bianchi identity we can reformulate the first integrability condition (1.7a) denotes the Ricci contraction. Notice that E := R * R is symmetric, but does in general not satisfy the Bianchi identity. A proof of the following lemma can be found in [ST69].
Applying this to the symmetric endomorphism in (3.1) and using (2.2c) we get This shows: Proposition 3.2. The first algebraic integrability condition for an algebraic curvature tensor R = W + T with Weyl part W and (not necessarily trace free) Ricci part T is equivalent to In particular, an algebraic curvature tensor satisfying the first algebraic integrability condition is aligned.
As a consequence of Proposition 2.6 we get: Corollary 3.3. A diagonalisable algebraic curvature tensor satisfies the first algebraic integrability condition.

3.2.
Integrability implies diagonalisability. The aim of this subsection is to prove Theorem 1.4. In view of Propositions 2.6 and 3.2 it is sufficient to show that if an algebraic curvature tensor satisfies the algebraic integrability conditions, then W + and W − have the same characteristic polynomial. We will first prove that the first algebraic integrability condition implies that W + has the same characteristic polynomial either as +W − or as −W − . We then prove that the latter contradicts the second algebraic integrability condition. We saw that an algebraic curvature tensor which satisfies the first algebraic integrability condition is aligned and thus of the form (2.10) in a suitable orthogonal basis. The first integrability condition in the form (3.3) then translates to for some r 0 and If we regard (w +1 , w +2 , w +3 ) and (w −1 , w −2 , w −3 ) as vectors in R 3 , then each equation in (2.10b) describes the plane through the origin with normal (1, 1, 1) and each equation in (3.4a) the sphere of radius r centered at the origin. Hence the solutions to both equations lie on a circle and can be parametrised in polar coordinates by a radius r 0 and angles ϕ + , ϕ − as Assume now that W + and W − have different eigenvalues. Then we have T ± = 0. Indeed, if t α = 0 for some α, then (3.4b) shows that w +α = w −α for this index α. But then (3.5) implies that W + and W − have the same eigenvalues, which contradicts our assumption. Hence T ± = 0.
Without loss of generality we will set s = 0 in what follows, because integrability is independent of the scalar curvature s. This follows from the fact that the Killing tensor corresponding to the scalar component in the Ricci decomposition is proportional to the metric and thus to the identity when regarded as an endomorphism. But adding a multiple of the identity only alters the eigenvalues, not the eigendirections, and therefore has no effect on integrability.
After these considerations, the independent components (2.11) of our algebraic curvature tensor are Choosing a 1 = b 1 = c 1 = d 1 = 0 in the second integrability condition (1.7b) for this algebraic curvature tensor yields since only terms with i = j = k = l = 0 contribute to the contraction. For (a 2 , b 2 , c 2 , d 2 ) = (0, 1, 2, 3) this is gives where the sums run over the three cyclic permutations (α, β, γ) of (1, 2, 3). We conclude that α=1,2,3 Substituting (3.5) into this equation yields, after some trigonometry, Substituting the solutions The first case is excluded by assumption and we conclude that W + and −W − have the same eigenvalues. Consequently we can choose an orthonormal basis of V such that Then the only remaining independent components of the algebraic curvature tensor (3.6) are R 0αβγ = w α . (3.8) Now consider the second integrability condition (1.7b) for (a 2 , b 2 , c 2 , d 2 ) = (0, 1, 2, 3), written as The symmetries then imply that the only independent components of (3.10) are Now consider the full contraction of (3.9): Since the Ricci tensor of (3.8) is zero, this is results in Using the symmetries (3.11) of E ijkl and those of R ijkl , we can always permute a particular index to the first position: And since both E ijkl and R ijkl vanish unless {i, j, k, l} = {0, 1, 2, 3}, we get Here again, the sums run over the three cyclic permutations (α, β, γ) of (1, 2, 3). Substituting (3.8) and (3.12) yields
3.3. Solution of the second integrability condition. It remains to prove Theorem 1.5. In the preceding subsections we showed that an algebraic curvature tensor which satisfies the algebraic integrability conditions is diagonal in a suitable orthogonal basis of V and that any diagonal algebraic curvature tensor satisfies the first of the two algebraic integrability conditions. We therefore now solve the second integrability condition (1.7b) for a diagonal algebraic curvature tensor in the form (2.13). To this aim consider the tensor appearing on the left hand side of (1.7b). Suppose it does not vanish. Then we have b 1 ∈ {a 2 , b 2 } and d 1 ∈ {c 2 , d 2 }, so without loss of generality we can assume b 1 = b 2 and d 1 = d 2 . Then only terms with i = j = a 2 and k = l = c 2 contribute to the contraction and hence {a 1 , a 2 } = {c 1 , c 2 }. If now a 2 = c 2 , then (3.13) vanishes under complete antisymmetrisation in a 2 , b 2 , c 2 , d 2 . This means that the left hand side of (1.7b) vanishes unless Note that a 2 , b 2 , c 2 , d 2 and therefore a 1 , b 1 , c 1 , d 1 have to be pairwise different due to the antisymmetrisation.
For dim V = 4 in particular, (1.7b) reduces to a sole condition, which can be written as (3.14) The form of this equation allows us to give an isometry invariant reformulation of the algebraic integrability conditions, which does not rely on diagonalising the algebraic curvature tensor. This also entails the isometry invariants in Propositions 1.6.
Proposition 3.4. The algebraic integrability conditions for an algebraic curvature tensor with Weyl part W and (not necessarily trace free) Ricci part T are equivalent to and the linear dependence of W, W 2 − T 2 and the identity matrix.
Proof. This is a consequence of Theorem 1.4 and Proposition 2.6, together with Corollary 3.3 and the second integrability condition in the form (3.14). Indeed, (3.15a) is the definition of alignedness for R and tr( * W k ) = 0 is equivalent to tr W k + = tr W k − . For k = 1 this equation is the Bianchi identity in the form (2.2d). Hence (3.15b) means that W + and W − have the same characteristic polynomial. The linear dependence of W, W 2 − T 2 and the identity matrix is equivalent to (3.14).
For t 1 = t 2 = t 3 = 0 the left hand side of (3.14) is the Vandermonde determinant. Therefore This proves the first part of Theorem 1.5, for this equation can be written in the form The second part follows from the residual action of the isometry on diagonal algebraic curvature tensors as described in Subsection 2.5. Regarding the eigenvalues w 1 , w 2 , w 3 ∈ R as parameters subject to the restriction we can solve this equation for t 1 , t 2 , t 3 ∈ R, depending on the number of equal eigenvalues: (ii) If w α = w β = w γ , then (3.17) implies that the solutions of (3.16) are those are equivalent under the isometry group, we can assume t β = t α without loss of generality. (iii) If w 1 = w 2 = w 3 , then (3.17) implies that w 1 = w 2 = w 3 = 0. In this case (3.16) is satisfied for any choice of t 1 , t 2 , t 3 ∈ R. In other words, an algebraic curvature tensor with zero Weyl component is always integrable.

The algebraic geometry of the KS-variety
We have seen that under isometries any integrable Killing tensor is equivalent to one whose algebraic curvature tensor is diagonal and that the space of such tensors is described by the following variety.
Definition 4.1. Let A ∼ = R 5 be the vector space of matrices whose symmetric part is diagonal and trace free, i. e. of matrices of the form We denote by V ⊂ A the algebraic variety defined by the equation The Killing-Stäckel variety P V is a 3-dimensional projective subvariety in P 4 and the quotient P V /S 4 encodes all information on integrable Killing tensors modulo isometries, the metric and scalar multiples. In this section we will investigate its structure from a purely algebraic geometric point of view. This will finally lead to a complete and explicit algebraic description of Stäckel systems.
Recall that Stäckel systems are n-dimensional vector spaces of mutually commuting integrable Killing tensors. Note that every Stäckel system contains the metric and that the metric corresponds to the zero KS-matrix. Therefore, if we assume that the algebraic curvature tensors of the Killing tensors in a Stäckel system are mutually diagonalisable, then Stäckel systems on S 3 correspond to 2-dimensional planes in V or projective lines in the projectivisation P V. We will see a posteriori that our assumption is justified. Hence Stäckel systems constitute a subvariety of the variety of projective lines on P V, also called the Fano variety of P V and denoted by F 1 (P V).
Let D 3 be the generic determinantal variety. This is the variety of 3 × 3 matrices with vanishing determinant or, equivalently, matrices of rank one or two. By definition P V is a subvariety of PD 3 . Hence its Fano variety F 1 (P V) is a subvariety of the Fano variety F 1 (PD 3 ). But the latter is well understood. It contains different types of projective linear spaces: (i) projective spaces of matrices whose kernel contains a fixed line, (ii) projective spaces of matrices whose image is contained in a fixed plane and (iii) the projective plane of antisymmetric matrices.
This motivates why we seek for such spaces in P V. Note that the projective plane of antisymmetric matrices is obviously contained in P V.
Definition 4.3. We call a projective subspace in the KS-variety consisting of matrices whose kernel contains a fixed line (respectively whose image is contained in a fixed plane) an isokernel space (respectively an isoimage space). The adjugate of a matrix M ∈ V is The adjugate matrix of a rank two 3 × 3 matrix has rank one, which means that the columns of Adj M span a line in R 3 . And because det M = 0, we deduce from (4.2) that the kernel of M is the column space of Adj M . The adjugate matrix of a rank one 3 × 3 matrix is zero. It is known, that the singular locus of PD 3 is the subvariety of rank one matrices. The following proposition characterises the singular locus of the KS-variety. For simplicity we will be a bit imprecise and not strictly distinguish between a matrix M ∈ V \{0} and its image RM ∈ P V under the canonical projection V \{0} → P V. Depending on the context, we will refer to an element of P V as "matrix" or "point".
Proposition 4.4 (Singularities of the KS-variety). The KS-variety P V has ten singular points, given by the six rank one matrices and the four skew symmetric matrices where (α, β, γ) denotes an even permutation of (1, 2, 3). Moreover, the rank one matrices in P V are exactly those six in (4.4a).
Definition 4.5. We will call the six singularities V ±α the rank one singular points in P V and the four singularities C i the skew symmetric singular points in P V.
Proof. P V is singular at RM if and only if V is singular at M . Since V is the zero locus of the determinant function det : A → R, this is the case if and only if the derivative of the determinant function at M ∈ V, given by is the zero map. The condition that tr(A Adj M ) = 0 for all A ∈ A is equivalent to We leave it to the reader to verify that the solutions M ∈ V of these equations are exactly the matrices (4.4) and their multiples. The last statement now follows from (4.5) and the equivalence of rank M = 1 and Adj M = 0 for 3 × 3-matrices.
Remark 4.6. Under the natural S 4 -action on P V, all rank one singular points are equivalent. This means that the singularieties V ±α are all mapped to a single point V in the quotient P V /S 4 . The same holds for the skew symmetric singular points, which are mapped to a single point C ∈ P V /S 4 . We now compute the isokernel spaces in P V. To find all matrices M ∈ V annihilating a given vector n ∈ R 3 , we consider the equation M n = 0 as an equation in M for a fixed n = (n 1 , n 2 , n 3 ), where M is of the form M = ∆ + T with We can regard t = (t 1 , t 2 , t 3 ) ∈ R 3 as a parameter and solve ∆ n = −T n for ∆.
For the time being, let us assume n 1 n 2 n 3 = 0. We then obtain a linear family of matrices All these matrices satisfy det M = 0 and the condition tr M = 0 imposes the restriction on the parameter t ∈ R 3 . This shows that the matrices in V annihilating a fixed vector n = 0 form a linear subspace. If the right hand side of (4.6b) is zero, i. e. if the condition is void, this subspace has dimension three and defines a projective plane in P V. This happens if and only if If the right hand side of (4.6b) is not zero, this subspace has dimension two and defines a projective line in P V. One can check that this also holds true for the case n 1 n 2 n 3 = 0. We have shown the following: Lemma 4.7. If n ∈ R 3 satisfies (4.7), then the set of matrices in P V annihilating n is a projective plane of the form (4.6a). Otherwise it is a projective line.
Recall that the kernel of a rank two matrix of the form (4.1) is the row space of the adjugate matrix (4.3). This defines a rational map π : P V P 2 whose fibres are the maximal isokernel spaces. This map is well defined except for the six rank one matrices in P V. We want to give a parametrisation of the isokernel lines that is uniform and more geometric than (4.6). For this purpose we define two embeddings ι, ν : P 2 → P V given by −n 1 n 2 n 3 n 1 n 1 n 2 n 2 3 − n 2 1 −n 2 n 3 −n 3 n 1 n 2 n 3 n 2 1 − n 2 2    (4.8) for n = (n 1 : n 2 : n 3 ) ∈ P 2 . One readily checks that ι(n) and ν(n) indeed lie in P V. Note that the image of ι is the projective plane of skew symmetric matrices in P V. Under the inclusion P V ⊂ PA ∼ = P 4 the map ι is a linear embedding of P 2 in P 4 whilst ν is the composition of the Veronese embedding P 2 → P 5 (n 1 : n 2 : n 3 ) → n 2 1 : n 2 2 : n 2 3 : n 2 n 3 : n 3 n 1 : n 1 n 2 and a projection P 5 P 4 under which the Veronese surface remains smooth.
Proposition 4.8 (Isokernel spaces in the KS-variety). Recall from Proposition 4.4 that the singular points in the KS-variety P V are the four skew symmetric singular points C 0 , C 1 , C 2 , C 3 as well as six rank one singular points V ±α , α = 1, 2, 3.
(i) The projective plane through V +1 , V +2 and V +3 is an isokernel plane in P V and containes C 0 . In the same way, the three points V −α , V +β and V +γ define an isokernel plane through C α for α = 1, 2, 3.
Let the maps π : P V P 2 and ι, ν : P 2 → P V be defined as above. (ii) For any non-singular point M ∈ P V, the points M , ι(π(M )) and ν(π(M )) are collinear but do not all three coincide. The projective line they define lies in P V and is an isokernel space through M . Each of these lines cointains a unique skew symmetric point, namely ι(π(M )).
Moreover, any isokernel space in P V of maximal dimension is of either of the above forms and hence contains a unique skew symmetric point.
Remark 4.9. A corresponding characterisation of isoimage spaces follows from the fact that matrix transposition defines an involution of P V which interchanges isokernel and isoimage spaces.
Proof. (i) follows directly from Lemma 4.7 and the explicit form (4.4) of the singular points. For (ii) notice that n := π(M ) is the kernel of M and well defined for M = V α . The definitions (4.8) of ι and ν then show that n is also the kernel of ι(n) and ν(n). These definitions also show that M , ι(n) and ν(n) do not all coincide for M = C i . Their collinearity follows from Lemma 4.7 in the case where n does not satisfy (4.7) and from ι(n) = ν(n) in case it does. The last statement is now a consequence of Lemma 4.7.
We can illustrate the content of Proposition 4.8 geometrically as follows. This is depicted in Figure 1. The six rank one singular points of P V ⊂ P 4 constitute the vertices of a regular octahedron in P 4 whose faces (and the planes they define) are entirely contained in P V. The set of eight faces is divided into two sets of nonadjacent faces, corresponding to the four isokernel planes (shaded) respectively the four isoimage planes in P V. Opposite faces intersect in their respective centres. These are the skew symmetric singular points of P V and the intersection points of the octahedron with the projective plane of skew symmetric matrices. In Figure 1 this ninth projective plane in P V is depicted as the insphere of the octahedron.
The permutation group S 4 acts on this configuration by symmetries of the octahedron preserving the two sets of non-adjacent faces. Matrix transposition corresponds to a point reflection, which exchanges opposite faces and completes the S 4 -action to the full octahedral symmetry.
In anticipation of the results in the next section we introduce the following name:

Stäckel systems
In this section we express the condition that two Killing tensors commute as an algebraic condition on the corresponding algebraic curvature tensors. This will eventually justify the term "Stäckel lines" chosen in Definition 4.10 by proving that these lines indeed correspond to Stäckel systems.
Remark 5.1. Most results in this and the following section are valid for an arbitrary non-flat constant curvature (pseudo-)Riemannian manifold M that is embedded into some vector space V as the hypersurface of length one vectors with respect to a non-degenerated symmetric bilinear form g on V . Note that any non-flat constant curvature (psudo-)Riemannian manifold is locally isometric to such a model. 5.1. Young tableaux. Throughout this section we use Young tableaux in order to characterise index symmetries of tensors. Young tableaux define elements in the group algebra of the permutation group S d . That is, a Young tableau stands for a (formal) linear combination of permutations of d objects. In our case, these objects will be certain tensor indices. For the sake of simplicity of notation we will identify a Young tableau with the group algebra element it defines. It will be sufficient for our purposes to consider the examples below. For more examples and a detailed explanation of the notation as well as the techniques used here, we refer the reader to [Sch12].
A Young tableau consisting of a single row denotes the sum of all permutations of the indices in this row. For example, using cycle notation, a 2 c 1 c 2 = e + (a 2 c 1 ) + (c 1 c 2 ) + (c 2 a 2 ) + (a 2 c 1 c 2 ) + (c 2 c 1 a 2 ) . This is an element in the group algebra of the group of permutations of the indices a 2 , c 1 and c 2 (or any superset). In the same way a Young tableau consisting of a single column denotes the signed sum of all permutations of the indices in this column, the sign being the sign of the permutation. For example, We call these row symmetrisers respectively column antisymmetrisers. The reason we define them without the usual normalisation factors is that then all numerical constants appear explicitly in our computations (although irrelevant for our concerns).
The group multiplication extends linearly to a natural product in the group algebra. A general Young tableau is then simply the product of all column antisymmetrisers and all row symmetrisers of the tableau. We will only deal with Young tableaux having a "hook shape", such as the following: (5.1) The inversion of group elements extends linearly to an involution of the group algebra. If we consider elements in the group algebra as linear operators on the group algebra itself, this involution is the adjoint with respect to the natural inner product on the group algebra, given by defining the group elements to be an orthonormal basis. Since this operation leaves symmetrisers and antisymmetrisers invariant, it simply exchanges the order of symmetrisers and antisymmetrisers in a Young tableau. The adjoint of (5.1) for example is Properly scaled, Young tableaux define projectors onto irreducible S d -representations.
We are now able to state a lemma which expresses the decomposition given by the Littlewood-Richardson rule, in terms of Young projectors. Its proof can be found in [Sch12].
Lemma 5.2. In the group algebra of the permutation group of the indices a 1 , a 2 , c 1 , c 2 , b 1 and d 2 the following identity holds: The action of the permutation group on tensors extends linearly to an action of the group algebra. In particular, any Young tableau acts on tensors with corresponding indices. For example, To give another example, the operator (5.1) acts on a tensor T b1b2d1d2a2c2 by an antisymmetrisation in the indices b 1 , a 2 , c 2 and a subsequent symmetrisation in the indices b 1 , b 2 , d 1 , d 2 . In the same way its adjoint (5.2) acts by first symmetrising and then antisymmetrising.

Commuting Killing tensors.
In the following we express the fact that two Killing tensors commute (as endomorphisms) as a purely algebraic condition on their corresponding algebraic curvature tensors. Notice that the commuting of Killing tensors as endomorphisms on the tangent space is not equivalent to the commuting of their respective algebraic curvature tensors as endomorphisms on Λ 2 V , as one may guess naïvely.
Proposition 5.3 (Algebraic counterpart of the commutator). Let K andK be two Killing tensors on a non-flat constant curvature manifold with algebraic curvature tensors R andR. Then the following statements are equivalent Proof. Using ∇ v x b = v b we write (1.6) in local coordinates for coordinate vectors v = ∂ α and w = ∂ β : The product of K andK, regarded as endomorphisms, is then given by In [Sch12] we proved the following identity: As a consequence of the antisymmetry of algebraic curvature tensors in the last index pair, the term x b2 x d1 does not contribute when substituting this identity into the previous expression: The commutator [K,K] is therefore given by and vanishes if and only if We can drop the restriction g(x, x) = 1 since RM ⊆ V is open. We can also drop the restrictions g(x, v) = 0 and g(x, w) = 0 by decomposing arbitrary vectors v, w ∈ V under the decomposition V = T x M ⊕ Rx. To see this, notice that (5.5) is trivially satisfied for v = x or for w = x. Indeed, in this case the tensor R a1b1a2b2Rc1d1c2d2 (5.7) is implicitely symmetrised over five indices and Dirichlet's drawer principle tells us that this comprises a symmetrisation in one of the four antisymmetric index pairs. This means we can omit the restrictions (5.6) completely. In other words, [K,K] = 0 is equivalent to (5.5) being satisfied for all x, v, w ∈ V . Now notice that the tensor appearing in (5.5) is completely symmetric in the indices a 1 , a 2 , c 1 and c 2 and completely antisymmetric in the indices b 1 and d 2 . Applying (5.3) to this tensor therefore yields (5.8) From the definition of the Young tableaux and their adjoints, together with the properties we see that (5.8) simplifies to When substituted into (5.5), this yields 1 72 As elements of the group algebra of the permutation group, the Young tableaux are linear combinations of permutations π of the indices a 1 , a 2 , c 1 , c 2 , b 1 , d 2 . For any such permutation π we have where π = π −1 is the adjoint of π. We can thus replace the Young tableaux in (5.9) acting on upper indices by its ajoint acting on lower indices: x a1 x a2 x c1 x c2 v b1 w d2 = 0 (5.10) Now notice that the second Young tableau involves a symmetrisation over the five indices b 1 , a 1 , a 2 , c 1 , c 2 and that, as above, the symmetrisation of the tensor (5.7) in any five indices is zero. Hence the second term in (5.10) vanishes and we obtain Recall that [K,K] = 0 is equivalent to this condition being satisfied for all x, v, w ∈ V . By polarising in x we get for all x, y, z, t, v, w ∈ V and hence This is the same as (5.4b) after appropriately renaming, lowering and rising indices. We have proven the equivalence (5.4a) ⇔ (5.4b).
We now prove the equivalence (5.4b) ⇔ (5.4c). Start from (5.4b) by expanding the Young tableau: In order to sum over all 4! permutations when carrying out the symmetrisation in the indices b 1 , b 2 , d 1 , d 2 , one can first take the sum over the 4 cyclic permutations of b 1 , b 2 , d 1 , d 2 , then fix the index b 1 and finally sum over all 3! permutations of the For a better readability we underlined each antisymmetrised index. Permuting the indices of the terms in the parenthesis under symmetrisation in b 2 , d 1 , d 2 and antisymmetrisation in b 1 , a 2 , c 2 we can gather the first and last as well as the second and third term: Using the symmetries of R b1a2b2 , the terms in the inner parentheses can be rewritten as As above, when carrying out the antisymmetrisation over b 1 , a 2 , c 2 , we can first sum over the three cyclic permutations of b 1 , a 2 , c 2 , then fix b 1 and finally sum over the two permutations of a 2 , c 2 . This results in The symmetrisation of this in b 1 , b 2 , d 1 , d 2 yields where we again underlined antisymmetrised indices. By the antisymmetry of algebraic curvature tensors in the second index pair, all but the fourth and fifth term in the parenthesis vanish and we get Due to the symmetrisation and antisymmetrisation we can permute the indices of the terms inside the parenthesis to find that Recall that this equation has been obtained from (5.4b) by a symmetrisation. This proves (5.4b) ⇒ (5.4c). The converse follows easily by antisymmetrising (5.4c) in b 1 , a 2 , c 2 . This achieves the proof of the theorem.
Remark 5.4. Obviously (5.4b) must be true forR = R. This fact is not evident, but has been proven in [Sch12].
For diagonal algebraic curvature tensors R andR on a four-dimensional vector space the condition (5.4c) can be written in the form for all distinct i, j, k ∈ {0, 1, 2, 3}.
Corollary 5.5. Stäckel lines 3 in the KS-variety correspond to Stäckel systems. More precisely, the preimage of a Stäckel line under the map from integrable Killing tensors with diagonal algebraic curvature tensor to the KS-variety is a Stäckel system. In particular, the Killing tensors in a Stäckel system have simultaneously diagonalisable algebraic curvature tensors.
Proof. A Stäckel line is the projective line through the (collinear but not coinciding) points M , ι(π(M )) and ν(π(M )) in the KS-variety for any non-singular KS-matrix M . We have to prove that corresponding Killing tensors K 0 , K 1 and K 2 mutually commute. We first show that K 1 and K 2 commute. Let R andR be their diagonal algebraic curvature tensors. From their KS-matrices (4.8) and ∆ α = w β − w γ we can read off the values w α = 0 t α = n αwα = n 2 αtα = n β n γ , where we have neglected the scalar curvature, which can be chosen appropriately. The diagonals of R andR are then given by the columns of the matrix  1 +n 1 n 2 1 + n 2 n 3 1 +n 2 n 2 2 + n 3 n 1 1 +n 3 n 2 3 + n 1 n 2 1 −n 1 n 2 1 − n 2 n 3 1 −n 2 n 2 2 − n 3 n 1 1 −n 3 n 2 3 − n 1 n 2 The four square matrices in (5.11) can be obtained from this matrix by discarding the rows (1, 2, 3), (1, 5, 6), (2, 4, 6) respectively (3, 4, 5) and it is not difficult to see that their determinants vanish. This shows that K 1 and K 2 commute. If ι(π(M )) and ν(π(M )) are linearly independent, then M is a linear combination of both and hence K 0 is a linear combination of g, K 1 and K 2 . In other words, K 0 , K 1 and K 2 mutually commute.
Consequently, the diagonal algebraic curvature tensors R andR of K 0 respectively K 2 are given by As above, they satisfy the commutation condition (5.11) and we conclude that K 0 , K 1 and K 2 commute.
It remains to prove that every Stäckel system consists of Killing tensors with mutually diagonalisable algebraic curvature tensors. This follows from the fact that every Stäckel system containes a Killing tensor with simple eigenvalues and that such a Killing tensor K uniquely determines the Stäckel system [Ben97]. By the very definition of a Stäckel system, K is integrable and by Theorem 1.4 we can assume it to have a diagonal algebraic curvature tensor. The corresponding point on the KS variety lies on a Stäckel line. By the above, K then lies in a Stäckel system with diagonal algebraic curvature tensors. This proves the statement, since the Stäckel system determined by K is unique.
We can now prove theorem Theorem 1.10. Recall from Proposition 4.8 that each Stäckel line contains a unique skew symmetric point ι(n), where n ∈ P 2 , and that each non-singular skew symmetric point ι(n) determines a unique Stäckel line, given by the two points ι(n) = ν(n). Moreover, the singular skew symmetric points are the four points n = (±1 : ±1 : ±1) ∈ P 2 for which ι(n) = ν(n). Hence it suffices to show that the subspace spanned by ι(n(t)) and ν(n(t)) has a well defined limit for t → 0 if n(0) is one of these and that this limit depends onṅ(0) ∈ P 2 . We leave it to the reader to verify that the limit space is spanned by ι(n(0)) and ι(ṅ(0)).

Geometric constructions of integrable Killing tensors
In this section we present several geometric constructions of integrable Killing tensors and interpret each of them within our algebraic picture from the last section.
6.1. Special Killing tensors. There are two ways in which integrable Killing tensors are related to geodesically equivalent metrics. The first is via special conformal Killing tensors.
Definition 6.1. A special conformal Killing tensor on a Riemannian manifold is a symmetric tensor L αβ satisfying the Sinjukov equation as can be seen from contracting α and β.
Special conformal Killing tensors parametrise geodesically equivalent metrics in the following way [Sin66].
Theorem 6.2. A metricg is geodesically equivalent to g if and only if the tensor is a special conformal Killing tensor.
From the definition we see that special conformal Killing tensors on a manifold M form a vector space which is invariant under the isometry group of M . In other words, they define a representation of this group. The following lemma shows that this representation is isomorphic to a subrepresentation of the representation of the isomorphism group on Killing tensors. Lemma 6.3. If L is a special conformal Killing tensor, then K := L − (tr L)g is a Killing tensor. This defines an injective map from the space of special conformal Killing tensors to the space of Killing tensors, which is equivariant with respect to the action of the isometry group.
Proof. It is straightforward to check the first statement. Taking the trace on both sides of K = L − (tr L)g yields tr K = (1 − n) tr L and hence This shows injectivity. Equivariance is obvious.
Definition 6.4. We define a special Killing tensor to be a Killing tensor of the form where L is a special conformal Killing tensor.
Proposition 6.5. A special conformal Killing tensor as well as the corresponding special Killing tensor have vanishing Nijenhuis torsion. In particular, special Killing tensors are integrable.
Proof. Substituting the Sinjukov equation (6.1) into the expression (1.3) for K = L shows that the Nijenhuis torsion of L is zero. Substituting K = L − (tr L)g into (1.3) shows that this implies that the Nijenhuis torsion of K is also zero.
For a non-flat constant curvature manifold M ⊂ V we can identify the subrepresentation of special Killing tensors inside the space of Killing tensors. Recall that the space of Killing tensors is isomorphic to the space of algebraic curvature tensors on V and an irreducible GL(V )-representation. Under the subgroup SO(V ) ⊂ GL(V ) this space decomposes into a Weyl component and a Ricci component.
Definition 6.6. The Kulkarni-Nomizu product of two symmetric tensors h and k on a vector space V is the algebraic curvature tensor h k, given by (6.4) The Ricci component consists of all algebraic curvature tensors of the form h g, where h is a symmetric tensor. Therefore the dimension of the Ricci part is equal to N (N + 1)/2 where N = dim V . This is exactly the dimension of the space of special conformal Killing tensors on M [Sin66]. But there is no other SO(V )subrepresentation of the same dimension inside the space of algebraic curvature tensors. This shows the following.
Lemma 6.7. A special Killing tensor on a non-flat constant curvature manifold is a Killing tensor whose algebraic curvature tensor R has a vanishing Weyl tensor. That is, R is of the form R = h g for some symmetric tensor h on V , where " " denotes the Kulkarni-Nomizu product (6.4).
Alternatively, this lemma can be checked directly by verifying that (6.3) satisfies (6.1) if K has an algebraic curvature tensor of the form R = h g.
The following consequence is a special case of the so called "cone construction" [MM10]: Any special conformal Killing tensor on a Riemannian manifold M can be extended to a covariantly constant symmetric tensor on the metric cone over M . In our case, where M ⊂ V is a constant curvature manifold, this cone is nothing but the embedding space V . That is why a special conformal Killing tensor is the restriction of a constant symmetric tensor on the ambient space.
Proposition 6.8. Let R = h g be the algebraic curvature tensor of a special Killing tensor K = L − (tr L)g on a non-flat constant curvature manifold M ⊂ V . Then the corresponding special conformal Killing tensor is the restriction of the (constant) symmetric tensor from V to M , where " Tr" denotes the trace on V and " tr" the trace on the tangent space. In particular, if h is tracefree, then L is simply the restriction of h from V to M .
Proof. The special Killing tensor K is given by substituting R = h g into (1.6) using (6.4). This yields for tangent vectors v, w ∈ T x M at a point x ∈ M , i. e. vectors x, v, w ∈ V with g(x, x) = 1 and g(x, v) = g(x, w) = 0. This simplifies to (6.8) Recall that "tr" denotes the trace on T x M , whereas "Tr" denotes the trace on V . Substituting (6.7) and (6.8) into the right hand side of (6.5) now yields (6.6).
For S 3 we can verify the integrability of a special Killing tensor on S 3 algebraically, using the Theorems 1.4 and 1.5: Lemma 6.9.
(i) R = h g is diagonalisable. More precisely, R is diagonal in a basis where h is diagonal. In this basis the diagonal elements of R are given in terms of the diagonal elements h i := h ii by (ii) If dim V = 4, the KS-matrix (1.9) of R = h g is antisymmetric with In particular it has determinant zero.
Corollary 6.10. The space of special Killing tensors on S 3 corresponds to the projective space of antisymmetric matrices inside the KS-variety.
Proposition 4.8 and Corollary 5.5 now give a unique representative in each Stäckel system.
Corollary 6.11. Every Stäckel system on S 3 contains a special Killing tensor which is unique up to multiplication with constants and addition of multiples of the metric.
6.2. Killing tensors of Benenti type. The second way in which integrable Killing tensors arise from geodesically equivalent metrics is described by the following theorem [LC04, Pai97, Ben92, MT98].
Theorem 6.12. If a metricg on a (pseudo-)Riemannian manifold is geodesically equivalent to g, then (6.9) is an integrable Killing tensor for g.
Definition 6.13. We will call a Killing tensor of the form (6.9) a Benenti-Killing tensor.
Recall that the standard metric on the unit sphere S n ⊂ V in a Euclidean vector space (V, g) is the restriction of the scalar product g from V to S n , which we denoted by g as well. Its geodesics are the great circles on S n . Consider the map for A ∈ GL(V ). Since A takes hyperplanes to hyperplanes, f takes great circles to great circles. Hence the pullbackg of g under f is geodesically equivalent to g by definition. The Killing tensor obtained from applying (6.9) tog is given by [Sch12] K x (v, w) = g(Ax, Ax)g(Av, Aw) − g(Ax, Av)g(Ax, Aw) Comparing with (1.6), we see that the algebraic curvature tensor of this Killing tensor is proportional to h h, where " " denotes the Kulkarni-Nomizu product (6.4) and h(v, w) := g(Av, Aw) = g(v, A t Aw) . Moreover, we can replace the scalar product h by any -not necessarily positive definite -symmetric tensor. This motivates the following definition.
Definition 6.14. We will say a Killing tensor on a constant curvature manifold of Benenti type if its algebraic curvature tensor is proportional to A Killing tensor of Benenti type with positive definite h is a Benenti-Killing tensor and thus integrable. With the aid of the algebraic integrability conditions one can verify that any Killing tensor of Benenti type is integrable [Sch12]. On S 3 we can check this directly, using the Theorems 1.4 and 1.5: Lemma 6.15.
(i) For a symmetric tensor h on a vector space V , the algebraic curvature tensor R = h h is diagonalisable. More precisely, R is diagonal in a basis where h is diagonal. In this basis the diagonal elements of R are given in terms of the diagonal elements h i := h ii by and has zero determinant and trace.
6.3. Benenti systems. Special Killing tensors and Benenti-Killing tensors only constitute particular cases of a more general construction of integrable Killing tensors out of special conformal Killing tensors. This construction appears in the literature in different guises and under a number of different names: "Newtonian systems of quasi-Lagrangian type" [RWML99], "systems admitting special conformal Killing tensors" [Cra03a], "cofactor systems" [Lun01], "bi-quasi-Hamiltonian systems" [CS02,Cra03b] or "L-systems" [Ben05]. Here we follow Bolsinov & Matveev, who call them "Benenti systems" [BM03], because their formulation is directly reflected in our algebraic description for S 3 . Recall that the adjugate matrix Adj M of a matrix M is the transpose of its cofactor matrix, which is equal to (det M )M −1 if M is invertible. Given a metric, we can identify endomorphisms and bilinear forms. This allows us to extend this definition to bilinear forms. As for endomorphisms, we denote the adjugate of a bilinear form L by Adj L, which is again a biliner form.
Theorem 6.16 ( [BM03]). For a special conformal Killing tensor L the family is a family of mutually commuting integrable Killing tensors. In particular, for λ = 0 we recover the Benenti-Killing tensor (6.9) from (6.2).
Definition 6.17. The family K(λ) = Adj(L − λg) of mutually commuting Killing tensors is called the Benenti system of the special conformal Killing tensor L.
In order to state the algebraic counterpart of the above theorem for constant curvature manifolds, we first need the following lemma.
Lemma 6.18. Let h be a symmetric tensor on a vector space V of dimension n+1. Then the algebraic curvature tensor is well defined and homogeneous in h of degree n − 1.
Proof. Chose a basis in which h is diagonal with diagonal elements h i := h ii . Then Adj h is also diagonal and has diagonal elements h 0 · · · h i−1 h i+1 · · · h n . By (6.10) the algebraic curvature tensor (6.11) is diagonal and given by This is well defined and homogeneous in h of degree n − 1.
Proposition 6.19 (Algebraic representation of Benenti systems). Let L be a special conformal Killing tensor on a non-flat constant curvature manifold M ⊂ V , i. e. the restriction of a (constant) symmetric tensorL from V to M . To avoid confusion, we denote the metric on V byĝ and its restriction to M ⊂ V by g.
Then the Killing tensors of the Benenti system correspond to the algebraic curvature tensors det(L − λĝ) . (6.13) In particular, a member of this Benenti system is of Benenti type unless λ is an eigenvalue ofL and it comes from a geodesically equivalent metric if −λ is sufficiently large.
Proof. For simplicity of notation, we absorb the trace terms "λg" and "λĝ" into L respectivelyL by formally setting λ to 0. At a fixed point x ∈ M we consider the block decomposition of the endomorphismL under the splitting V = T x M ⊕ Rx.
Using the blockwise LU decomposition ofL, we can compute the block decomposition of the adjugate ofL: Note that we have assumed without loss of generality that L is invertible. Now take a basis e 1 , . . . , e n of T x M and complete it with e 0 := x to a basis of V . With respect to this basis the components of the Killing tensor K corresponding to the algebraic curvature tensor (6.13) read That is, K is given by (6.12), as was to be shown.
The adjugate of an (n × n)-matrix M is a polynomial in M of degree n − 1, given by where χ M is the characteristic polynomial of M . Therefore a Benenti system is a polynomial in λ of degree n − 1: Up to signs, the leading coefficient of this polynomial is the metric, the next coefficient is the special Killing tensor (6.2) and the constant term is the Benenti-Killing tensor (6.9): Note that since all the K(λ) commute, the n coefficients K j span a vector space of mutually commuting integrable Killing tensors. If the n coefficients are linearly independent, then this space is a Stäckel system and the K j form a basis therein. However, it is in general an open question under which conditions this will be the case. For S 3 we can derive the answer directly from our algebraic description.
Lemma 6.20. For a symmetric tensor h on a four dimensional vector space we have (Adj h) (Adj h) det h = * (h h) * , (6.14) where * is the Hodge star operator.
Remark 6.21. Regarding (2.2), the conjugation of an algebraic curvature tensor with the Hodge star is simply a sign change in the trace free Ricci part: Proof. Choose a basis in which h is diagonal with diagonal elements h i := h ii . For simplicity of notation we assume that det h = 0. Then h i = 0 and Adj h is diagonal with diagonal elements det h/h i . Substituting Adj h for h in (6.10) and comparing to (6.10) shows that the diagonal algebraic curvature tensors for {i, j, k, l} = {0, 1, 2, 3}. Now the identity (6.14) follows from (2.13) and the fact that R and * R * differ in the sign of the trace free Ricci part.
Proposition 6.22. Let L be a special conformal Killing tensor on S 3 ⊂ V , i. e. the restriction of a (constant) symmetric tensorL from V to S 3 , and consider the corresponding Benenti system Then the coefficients K 0 , K 1 and K 2 span a Stäckel system if and only ifL has no triple eigenvalue.
Proof. By Proposition 6.19 and the Identity (6.14) the algebraic curvature tensor of K(λ) is the polynomial The coefficients R 0 , R 1 and R 2 are the algebraic curvature tensors of K 0 , K 1 and K 2 respectively. We can assume them to be diagonal. Since K 2 is minus the metric, K 0 , K 1 and K 2 are linearly independent if and only if R 0 and R 1 define two distinct points in the KS-variety. Choose a basis in whichL is diagonal with the eigenvalues Λ i on the diagonal. Without loss of generality we can assume thatL is trace free. Motivated by (2.12), we can thus parametriseL by a vector n ∈ R 3 as ThenL has a triple eigenvalue if and only if |n 1 | = |n 2 | = |n 3 |. This condition is equivalent to the condition that the KS-matrices ν(n) and ι(n) in (4.8) are proportional, i. e. define the same point in the KS-variety. Our proof will therefore be finished if we show that R 0 and R 1 define the points ν(n) respectively ι(n) in the KS-variety. That the algebraic curvature tensor R 1 = 2L g defines the point ι(n) is the statement of Lemma 6.9. And since R 0 = * (L L ) * differs fromL L in the sign of the trace free Ricci part, Lemma 6.15 implies that for R 0 the KS-matrix (1.9) is given by Now compare this to ν(n) in (4.8) to finish the proof.
The Benenti system of a special conformal Killing tensors L defines a map R → K(S 3 ) given by λ → K(λ) = Adj(L − λg). In the proof above we have actually shown that this defines a map from P 1 = R ∪ {∞} to the KS-variety which maps 0 → ν(n) and ∞ → ι(n), where n ∈ P 2 is determined by L. If the points ι(n) and ν(n) coincide, the image of this map is a single point. If they are distinct, the image is -by definition -the Stäckel line through both points. In this case the image of the Benenti system is this Stäckel line without the point ι(n).
We can now prove Theorem 1.8. It suffices to show that the image of diagonal algebraic curvature tensors of the form λ 2 h h is dense in the KS-variety. By Proposition 6.19 the algebraic curvature tensors of this form are dense in a Benenti system, so it suffices to show that the image of Benenti systems in the KS-variety is dense. But this follows from the preceding paragraph and part (ii) of Proposition 4.8.
6.4. Extension of integrable Killing tensors. If we fix an orthogonal decomposition R n1+n2+2 = R n1+1 ⊕ R n2+1 , then two algebraic curvature tensors R 1 on R n1+1 and R 2 on R n2+1 define an algebraic curvature tensor R 1 ⊕ R 2 on R n1+n2+2 . Correspondingly, two Killing tensors K 1 on S n1 and K 2 on S n2 define a Killing tensor K 1 ⊕ K 2 on S n1+n2+1 . In particular, we can extend any Killing tensor K on S n−1 by zero to a Killing tensor K ⊕ 0 on S n .
From the algebraic integrability conditions we see that K 1 ⊕ K 2 is integrable if K 1 and K 2 are both integrable. This direct sum operation defines an embedding where K(S n ) denotes the space of integrable Killing tensors on S n . In particular we have an embedding K(S n−1 ) → K(S n ) .
Definition 6.23. We will call (integrable) Killing tensors arising in the way described above extensions of Killing tensors from (S n1 , S n2 ) respectively S n−1 .
As an example, let us determine which integrable Killing tensors on S 3 are extensions from S 2 respectively (S 1 , S 1 ). Note that in dimension one and two all Killing tensors are integrable. This follows directly from the integrability conditions.
An algebraic curvature tensor R on R 4 is an extension by zero of an algebraic curvature tensor on R 3 under the inclusion (x 1 , x 2 , x 3 ) → (0, x 1 , x 2 , x 3 ) if and only if R a1b1a2b2 is zero whenever one of the indices is zero. In particular, the diagonal algebraic curvature tensor (1.8) on S 3 is an extension from S 2 if and only if w α + s 12 = −t α . In this case the KS-matrix (1.9) is equal to (4.6a) with n 1 = n 2 = n 3 . Therefore the extensions of Killing tensors from S 2 correspond to the isokernel planes in the KS-variety. In particular, the face centers of the octahedron in the KSvariety correspond to the extension of the metric on S 2 under the four embeddings S 2 ⊂ S 3 given by intersecting S 3 ⊂ R 4 with each of the four coordinate hyperplanes.
An algebraic curvature tensor R on R 4 is a sum under the decomposition of (x 0 , x 1 , x 2 , x 3 ) into (x 0 , x 1 ) and (x 2 , x 3 ) if and only if the components R a1b1a2b2 are zero unless all four indices are either in {0, 1} or in {2, 3}. In particular, a diagonal algebraic curvature tensor (1.8) is an extension from (S 1 , S 1 ) if and only if w 2 + s 12 = t 2 = w 3 + s 12 = t 3 = 0. In this case the KS-matrix (1.9) is equal to  where V +1 and V −1 are rank one singular points in the KS-variety, c. f. (4.4). Therefore the extensions of Killing tensors from (S 1 , S 1 ) correspond to the projective lines joining opposite vertices of the octahedron in the KS-variety. In particular, each vertex of the octahedron corresponds to an extension of the metrics on two orthogonal copies of S 1 in S 3 , given by intersecting S 3 ⊂ R 4 with a pair of orthogonal coordinate planes.

Separation coordinates
We eventually demonstrate how separation coordinates on S 3 and their classification arise naturally and in a purely algebraic way from our description of integrable Killing tensors.
Kalnins and Miller proved that separation coordinates on S n are always orthogonal [KM86]. From the work of Eisenhart we know that orthogonal separation coordinates are in bijective correspondence with Stäckel systems [Eis34]. And in the preceding sections we have proven that on S 3 every Stäckel system contains a special Killing tensor which is essentially unique. By definition, special Killing tensors are in bijective correspondence with special conformal Killing tensors. But Crampin has shown that the eigenvalues of a special conformal Killing tensor are constant on the coordinate hypersurfaces of the corresponding separation coordinates [Cra03a]. Hence, if these eigenvalues are simple and non-constant, they can be used as separation coordinates. This reduces the classification of separation coordinates on S 3 to a computation of the eigenvalues of special conformal Killing tensors. The computation is considerably simplified by the fact that a special conformal Killing tensor on S n ⊂ V is the restriction of a (constant) symmetric tensor on V .
7.1. Eigenvalues of special conformal Killing tensors. Let L be a special conformal Killing tensor on a non-flat constant curvature manifold M ⊂ V , i. e. the restriction from V to M of a (constant) symmetric tensorL. From now on we consider L andL as endomorphisms. We want to compute the eigenvalues λ 1 (x) . . . λ n (x) of L : T x M → T x M for a fixed point x ∈ M and relate them to the (constant) eigenvalues Λ 0 . . . Λ n ofL : V → V .
Let P : V → V be the orthogonal projection from V ∼ = T x M ⊕ Rx to T x M . Then the restriction of PLP to T x M is L and the restriction of PLP to x is zero. Therefore x is an eigenvector of PLP with eigenvalue 0 and the remaining eigenvectors and eigenvalues are those of L. This means the eigenvalue equation Lv = λv is equivalent to PLP v = λv for v ⊥ x. By the definition of P we have P v = v and PLv =Lv − g(Lv, x)x. Hence we seek common solutions to the two equations If λ is not an eigenvalue ofL, then (L − λ) is invertible and v is proportional to (L − λ) −1 x. The condition v ⊥ x then yields the equation for λ. In an eigenbasis ofL this equation reads That is, the zeroes of the function q(λ) are eigenvalues of L. As depicted in Figure 2, q(λ) goes to 0 for λ → ±∞, has poles at the eigenvalues Λ k ofL unless x k = 0 and is monotonely increasing in between. Let us first suppose thatL has only simple eigenvalues. In this case (7.1) is the defining equation for elliptic coordinates λ 1 (x), . . . , λ n (x) on S n with parameters Λ 0 < Λ 1 < · · · < Λ n [Neu59]. If x k = 0 for all k = 0, . . . , n, then q has exactly λ q(λ) Figure 2. Eigenvalues of a special conformal Killing tensor on S n .
n + 1 poles at Λ 0 , . . . , Λ n and n zeroes in between. If x k = 0, then q has no pole at Λ k . But x k = 0 means that the k-th eigenvector ofL is orthogonal to x and hence also an eigenvector of L with eigenvalue Λ k . We conclude that the eigenvectors v k ofL which are orthogonal to x are eigenvectors of L with the same eigenvalue Λ k , and that the remaining eigenvalues of L are the zeroes λ k of the function q(λ) with corresponding eigenvector v k = (L − λ k ) −1 x. In particular, the eigenvalues ofL and L intertwine: Note that the common denominator of the fractions in the sum (7.1) is the characteristic polynomial χL(λ) ofL. Therefore the the function q is the quotient of the characteristic polynomials of L andL. IfL has a simple eigenvalue, then the corresponding eigenspace is one dimensional and has only a trivial intersection with a generic tangent space to M , because M ⊂ V is of codimension one. IfL has a double eigenvalue, then the corresponding eigenspace is a plane and has a one-dimensional intersection with a generic tangent space to M . This defines a field of common eigenvectors for L andL on M . The corresponding eigenvalue is the same for L andL and hence constant on M . This also follows from (7.2). In the same way an eigenvalue ofL with multiplicity m + 1 defines an m-dimensional eigenspace distribution of L with constant eigenvalue. 7.2. Killing tensors and separation coordinates on S 2 . Before we turn to S 3 , let us briefly discuss integrable Killing tensors and separation coordinates on S 2 . The reason is that their extensions will appear on S 3 . First note that in dimension two every Killing tensor is integrable. So we do not have to deal with the integrability conditions. Note also that algebraic curvature tensors on R 3 have a zero Weyl part and are thus determined by their Ricci tensor T alone. This means that Killing tensors on S 2 ⊂ R 3 correspond to symmetric tensors T on R 3 , where the metric corresponds to the identity.
Stäckel systems on S 2 are of dimension two and contain the metric. That implies that each Stäckel system on S 2 corresponds to a plane of Ricci tensors spanned by the identity and some trace free symmetric tensor, the latter being unique up to multiples. Moreover, the isometry group acts by conjugation on the Ricci tensor. Accordingly, the classification of separation coordinates on S 2 modulo isometries is equivalent to the classification of trace free symmetric tensors T on R 3 under the orthogonal group O(3).
In particular, every Killing tensor on S 2 is special and we can make use of the results in the previous section. Depending on the multiplicities of the eigenvalues t 1 t 2 t 3 of the trace free Ricci tensor T of a Killing tensor on S 2 we consider the following two cases.
One of the eigenvalues only depends on x 1 and the other is constant: The curves of constant λ 1 (x) are circles of constant latitude and the meridians complete them to an orthogonal coordinate system -the spherical coordinates. The case t 1 = t 2 < t 3 is analogue and yields spherical coordinates as well. Note that even so, both coordinate systems are not isometric. This is because they differ not only by an isometry but also by an exchange of the two coordinates.
We can use the action of the isometry group O(3) to diagonalise the Ricci tensor and thereby the algebraic curvature tensor. Hence the projective variety of integrable Killing tensors on S 2 with diagonal algebraic curvature tensor is isomorphic to the projective plane P 2 , where the point (1 : 1 : 1) ∈ P 2 corresponds to the metric on S 2 . Moreover, Stäckel systems on S 2 correspond to projective lines through this point, since they always contain the metric. The residual isometry group action on Killing tensors with diagonal algebraic curvature tensor is the natural action of S 3 on P 2 by permutations. 7.3. The classification. We now recover the well known classification of separation coordinates on S 3 [Eis34,KM86]. In particular, this will explain the graphical procedure given in [KM86] in terms of the classification of trace free symmetric tensors under the orthogonal group. The results are summarised in Table 1. As before, we will denote by Λ 0 Λ 1 Λ 2 Λ 3 the ordered eigenvalues of the symmetric tensorL associated to a Stäckel system and parametrised by a vector (n 1 , n 2 , n 3 ) ∈ R 3 via (6.15a). Recall that we have the freedom to add a multiple of the metric to a Killing tensor or to multiply it with a constant. Both operations only affect the eigenvalues, not the eigenspaces, and therefore result in different parametrisations of the same coordinate system. Whereas Kalnins & Miller use this freedom to set Λ 0 = 0 and Λ 1 = 1, here we use it to assume thatL is trace free and then to consider (Λ 0 : Λ 1 : Λ 2 : Λ 3 ) respectively (n 1 : n 2 : n 3 ) as a point in P 2 .
(0123) Elliptic coordinates. We have already seen that if the eigenvalues ofL are simple, then the eigenvalues of L are also simple and define elliptic coordinates on S n given implicitely by (7.1).
(01(23)) Oblate Lamé rotational coordinates. IfL has a double eigenvalue Λ k−1 = Λ k , then the corresponding eigenvalue λ k of L is constant and can not be used as a coordinate. Nevertheless, we can derive the missing coordinate from the fact that the coordinate hypersurfaces are orthogonal to the eigenspaces of L. Let E be an eigenplane ofL and denote by E ⊥ its orthogonal complement in V . Recall that the corresponding eigenspace of L at a point x ∈ S n is given by the intersection of E with the tangent space T x S n . Its orthogonal complement in T x S n is the intersection of T x S n with the hyperplane that contains both x and E ⊥ . The coordinate hypersurfaces are therefore the intersections of hyperplanes containing E ⊥ with S n and can be parametrised by the polar angle in E. Hence this angle can be taken as a coordinate. By (7.2), L has single eigenvalues given by (7.1) if and only ifL has at most double eigenvalues and every double eigenvalue ofL determines a constant eigenvalue for L. If we replace every constant eigenvalue by the polar angle in the corresponding eigenplane ofL, we obtain a complete coordinate system. Concretely, in the case Λ 0 < Λ 1 < Λ 2 = Λ 3 the separation coordinates are defined by the solutions λ 1 (x) and λ 2 (x) of (7.1) together with the polar angle in the (x 2 , x 3 )-plane.
For Λ 0 < Λ 1 < Λ 2 = Λ 3 we have n 1 < n 2 = n 3 . The corresponding Stäckel line thus goes through the two points in the KS-variety given by the KS-matrices (4.8): Subtracting n 1 times the first matrix from the second, we see that the same Stäckel line is also determined by the two points (0(12)3) Prolate Lamé rotational coordinates. The case Λ 0 < Λ 1 = Λ 2 < Λ 3 is analogue and yields Lamé rotational coordinates which differ from the above by an isometry and an exchange of two of the coordinates.
We can take the non-constant eigenvalue λ 1 (x) as one separation coordinate on S 3 . However, since the other two eigenvalues of L are equal, we can not recover all separation coordinates from the special conformal Killing tensor L in this case.
On the other hand, the eigenvalue λ 1 (x) remains unchanged. Together, they define separation coordinates on S 3 . We denote this by writing the multiplicities ofL as (0(123)) respectively (0(1(23))), depending on whether the multiplicities of the Ricci tensor T are (123) or (1(23)). This yields the remaining two systems of separation coordinates: (0(1(23))) Spherical coordinates. These are extensions of spherical coordinates on S 2 and correspond to Stäckel lines through a face center and a vertex of the octahedron in the KS-variety.
(0(123)) Lamé subgroup reduction. These are extensions of elliptic coordinates on S 2 and correspond to Stäckel lines through a face center, but not a vertex of the octahedron in the KS-variety.
7.4. The classification space. The non-generic separation coordinates are characterised by n 1 = ±n 2 , n 2 = ±n 3 and n 3 = ±n 1 . This defines six projective lines that divide the projective plane of skew symmetric matrices in the KS-variety into twelve congruent triangles. The four singular skew symmetric points are given by |n 1 | = |n 2 | = |n 3 | and each of the triangles has two of them as vertices. Upon a blow-up in these points, the triangles become pentagons. Any of these pentagons constitutes a fundamental domain for the S 4 -action. Regarding our notation for the multiplicities ofL, we realise that the vertices are labeled by the different possibilities to parenthesise the ordered set of eigenvalues ofL with pairs of correctly matching parentheses and that the edges correspond to a single application of the associativity rule. This naturally identifies the classification space for separation coordinates on S 3 with the associahedron K 4 , also known as Stasheff polytope [Sta63].