Symmetry, Integrability and Geometry: Methods and Applications Superintegrable Extensions of Superintegrable Systems ⋆

A procedure to extend a superintegrable system into a new superintegrable one is systematically tested for the known systems on $\mathbb E^2$ and $\mathbb S^2$ and for a family of systems defined on constant curvature manifolds. The procedure results effective in many cases including Tremblay-Turbiner-Winternitz and three-particle Calogero systems.


Introduction
Given a natural Hamiltonian L with n degrees of freedom, satisfying some additional geometric conditions, it is shown in [1] how to generate a n + 1 degrees of freedom Hamiltonian H, called the extension of L and depending on an integer parameter m ∈ N \ {0}, such that H admits two new independent first integrals: H itself and a polynomial in the momenta of degree m. This implies that, if L is superintegrable with 2n−1 independent first integrals, then all the extended Hamiltonians H also are superintegrable for any m with the maximal number of 2n + 1 = 2(n + 1)−1 first integrals, one of them of arbitrary degree m whose expression is explicitly computed by means of a simple iterative process. The extension procedure, summarized in Section 2, is applied to the superintegrable systems on E 2 (Section 3) and S 2 (Section 5) as listed in [5]. These are all the superintegrable systems on S 2 and E 2 admitting two independent first integrals of degree two in the momenta other than the Hamiltonian. It is found that a great part of them admits superintegrable extensions and for some of them the extensions are explicitly determined. In Section 4 the possible natural Hamiltonians admitting an extension are determined on E n and the superintegrable systems of Calogero and Wolfes are considered as examples in E 3 . In Section 6 the extension procedure is applied to a class of Hamiltonians on constant curvature manifolds to which some generalizations of the Tremblay-Turbiner-Winternitz (TTW) system belong [10]. These generalizations of the TTW systems are superintegrable for rational values of certain parameters, for which they admit polynomial first integrals of degree greater than two [7,8]. The cases allowing the extensions are determined and the TTW systems are among them, obtaining in this way their superintegrable extensions. Other superintegrable generalizations in higher dimensions of the TTW system are obtained in [6].

Extensions of superintegrable systems
We resume in the following statement the main results proved in [1]. Theorem 1. Let Q be a n-dimensional (pseudo-)Riemannian manifold with metric tensor g.
X L is the Hamiltonian vector field of L and G(q i ), if and only if the following conditions hold: i) the functions G and V satisfy where ii) for c = 0 the extended Hamiltonian H is for c = 0 the extended Hamiltonian H is Dynamically, extended Hamiltonians (1) can be written as with H = h, and where constant η can be understood either as a separation constant, between the elements of H depending on (u, p u ) and those depending on (q i , p i ), or as a coupling constant merging together the Hamiltonian L on T * Q and the Hamiltonian depending on (u, p u ), to build the extended Hamiltonian H. Several examples are given in [1]. Under the hypotheses of Theorem 1, recalling that C κ (x) = d dx S κ (x), the polynomials in the momenta are first integrals of H of degree m. For example, where { , } are the Poisson brackets. Another way to calculate U m G is to apply the formula [2] where [·] denotes the integer part, D 1 = γ and Remark 1. For c = 0, if L 0 = 0 then p u is a first integral of the extended Hamiltonian and the extended potential is merely mAV , with m and A constants. Therefore the extension is trivial. Otherwise, a harmonic oscillator term in the variable u, attractive or repulsive, is added to the potential mAV . In the case c = 0, we remark that, in the extended Hamiltonian (5), the potential V is multiplied by a non-constant factor depending on u.
Remark 2. The expression (6) of γ(u) is determined in [1] as the general solution of the differential equation for real values of γ, u, c and κ. However, the general solution of equation (7) in the complex case can be considered too. Its expression is the same as (6) if we extend functions S κ and C κ to complex values of x and κ by using the standard exponential expressions of trigonometric functions. After this generalization, the extension procedure characterized by Theorem 1 can be applied also to the complex case.
In [1] it is proved that, if n > 1, then (2) admits a complete solution G depending on a maximal number of parameters (a i ), with i = 0, . . . , n, iff the sectional curvature of Q is constant and equal to mc. Once such a solution G is known, an extension of L is possible iff the compatibility condition (3) on V is satisfied. A sharper result on constant curvature manifolds is the following Proposition 1. On a manifold with constant curvature K the only eigenvalues mc for the Hessian equation (2) are either zero or the curvature K. Moreover, if K = 0 and mc = 0 then the only solutions of (2) are G = const.
Proof . The equation (2), written in components, becomes and the integrability conditions that each solution G(q i ) must satisfy are given by (see [1] for details) is the Riemann tensor of the metric. For a constant curvature manifold we have R hlij = K(g jl g hi − g il g hj ) and the above conditions become By choosing orthogonal coordinates, we see that, since i = j, the equations are identically satisfied for h = i, j. Otherwise, they reduce to (K − mc)g jj z i = 0. Hence, for mc = K, the only possibility is z i = 0 for all i (that is, G is a constant). For mc = K and mc = 0, by (2) we get G = 0, thus mc is not an eigenvalue.
In [1] it is shown that the first integral U m (G) is functionally independent from H, L and all its possible first integrals L i in T * Q. It is straightforward to see that L and L i are first integrals of H, therefore, if L is a superintegrable Hamiltonian with 2n − 1 first integrals, including L, then H is superintegrable too with 2n − 1 + 2 = 2(n + 1) − 1 first integrals including H itself. It follows that the extension procedure applied to a superintegrable Hamiltonian L, under the hypothesis of Theorem 1, always produces a new superintegrable Hamiltonian H. Given a superintegrable system of Hamiltonian L = 1 2 g ij p i p j + V (q i ) with a configuration manifold Q of constant curvature K, its extension to another superintegrable system of Hamiltonian H, when possible, can be obtained by applying the following algorithm (see also [1]): 1. Solve equation (2) on the manifold Q, with c = K/m, to compute the general form of the function G(q i , a 0 , . . . , a n ). (3) with the given V for some of the parameters (a i ) in G. This is a crucial step, because if no solution is found, except for the trivial one G = 0, then the extension is not possible.

Solve equation
3. Determine the extension via Theorem 1.
4. Compute U m (G) to obtain the additional first integral.
In the following sections we analyze several examples of extensions of superintegrable systems.

Superintegrable extensions of E 2 systems
Since the curvature of E 2 is zero, equation (2) admits, by Proposition 1, solutions G = 0 only for mc = 0. Thus, we are in the case c = 0 of Theorem 1 and the extended Hamiltonian H is in the form (4). In [1] the complete solution G of (2) is computed, in standard Cartesian coordinates of E 2 , as Equation (3) becomes here with the constraint a 0 = a 1 x 0 + a 2 y 0 , where x 0 , y 0 ∈ R or C and F is any regular function of the argument. The extension, after the non restrictive assumptions V 0 = u 0 = 0 and A = m −1 , becomes In [5] the list of all superintegrable potentials in E 2 with three independent quadratic in the momenta first integrals is given, up to isometries and reflections. In that article E 2 is assumed to be a two-dimensional complex manifold. According to Remark 2 we apply in this case the same procedure developed for the real case and allow all functions, variables and parameters (except for m ∈ N) to take indifferently real or complex values, in this one and all the following sections. By setting z = x + iy,z = x − iy (remark that, despite the notation, if x and y are complex coordinates, then z andz are not complex conjugate one of the other), the list is The equation of compatibility (3) is equivalent to a linear homogeneous expression in (a i ) with coefficients linear but not homogeneous in α i . This expression vanishes only for some suitable choices of the parameters a i and α i , depending on the extension parameters m, L 0 . The solution (9) of (3) shows that a non null function G of the form (8) satisfying (3) exists only for the following potentials V G particular cases of The potentials admitting G(a i ) depending on several a i allow the existence of different first integrals of the form U m G(a i ). However, since the system is already superintegrable by including a single U m G(a i ), all the other first integrals obtained in this way functionally depend on the known ones.
Because L 0 = 0, a necessary condition for the extensibility is the presence of a harmonic term in the potential V .

Extensions of the harmonic oscillators
As an example, we analyze into details the extensions of the isotropic harmonic oscillator (a particular case of E1, E3, E7, E8) and of the anisotropic one (a particular case of E2) In both cases we assume α 3 = 0 to avoid a trivial potential. The only possible extension for V i is with the constraint α 3 − mL 0 = 0 due to (3) that sets the potential in the tabulated form. The Hamiltonian (11) represents an anisotropic oscillator in E 3 and shows explicitly its superintegrability being m an integer. Recalling that U = p u − m −1 uX L , and by setting X L G = a 1 p x + a 2 p y = P , we obtain as an example the expression of U 4 G The anisotropic oscillator (10) admits, instead, two extensions, corresponding to the two different functions G and two different relations between α 3 and L 0 (items (iv) and (v) of the table with α 1 = α 2 = 0). In order to analyze this case in full generality, an iterated procedure of extension can be applied. By starting from the one-dimensional oscillator we build a first extension where we use x 2 = u and, because of (3), we have G 1 = a 1 x 1 and L 0 = ω/m 1 . The resulting potential is an anisotropic oscillator and our procedure produces the third first integral of degree m 1 in the momenta U m 1 G 1 = (p 2 − m −1 1 X H 1 ) m 1 G 1 . The potential of H 2 coincides with V a for m 1 = 2 and ω = 4α 3 (this is the unique case with a third quadratic first integral).
A further step is the research of an extension of H 2 where the value of L 0 has to be determined. In this case, since the general G 2 is (8), condition (3) becomes Hence, we get a 0 = 0, and If a 1 = 0, we need a 2 = 0 and If a 1 = 0 and a 2 = 0, then we have Finally, if both a 1 = 0 and a 2 = 0 the two conditions are satisfied iff m 2 1 = 1 (i.e., H 2 is the Hamiltonian of an isotropic oscillator) and In the first two cases, H 3 represents an anisotropic harmonic oscillator, which is superintegrable because m 1 , m 2 are integers, but they have different functions G 2 and consequently different first integrals If we set m 1 = 2, m 2 = m and ω = 4α 3 in order to restrict ourselves to the potential (10), we obtain two possible extensions: if a 1 = 0 we have the relation α 3 − mL 0 = 0 that gives the first form in the table. If, otherwise, a 2 = 0 we have the relation 4α 3 − mL 0 = 0 that gives the second one.
The extension procedure can be iterated indefinitely obtaining at the n-th step an n-dimensional anisotropic oscillator with a complete set of first integrals (U m 1 G 1 , U m 2 G 2 , . . . , U m n−1 G n ) of degree (m 1 , m 2 , . . . , m n−1 ), that, together with the n Hamiltonians (H 1 , H 2 , . . . , H n ) make the system superintegrable. We remark that the systems obtained in this way are characterized by the fact that the frequencies are all integer multiples of one of them. See [4,9] for additional details on superintegrability of anisotropic oscillators.

Superintegrable extensions of E n
It is straightforward to generalize to E n the procedure previously applied to E 2 . Let us consider in E n with Cartesian coordinates the Hamiltonian L = 1 2 p 2 1 + p 2 2 + · · · + p 2 n + V (x 1 , x 2 , . . . , x n ).
The general solution of (2) and (3) are G = a 0 + a 1 x 1 + a 2 x 2 + · · · + a n x n V = mL 0 x 1 + x 0 1 2 + · · · + x n + x 0 n 2 + F (a 1 x 2 − a 2 x 1 , . . . , a 1 x n − a n x 1 ), with the constraint a 0 = n i=1 a i x 0 i , where x 0 i ∈ R or C, F is any regular function of the arguments and L 0 = 0. The corresponding extension is We remark that, as well as in dimension 2, the presence of a harmonic term in V is a necessary condition for the extensibility.

Extensions of the three-body Calogero and Wolfes systems
We consider the particular case of n = 3. If a 2 = a 3 = a 1 and F (X 1 , X 2 ) in (12) is with k ∈ R, then coincides with the celebrated Calogero potential, which is a well known superintegrable system (see for example [3] and references therein). If, with the same choice for the a i , then F coincides with the Wolfes potential, a three-body superintegrable interaction whose dynamic equivalence with the Calogero potential is discussed in [3]. If L 0 = 0, then the first integrals of the extended Hamiltonian for m = 2, 3 and F in the Calogero form are respectively and P = X L (G) = p 1 + p 2 + p 3 is the conserved linear momentum.

Superintegrable extensions of S 2
In [1] the complete solution G of (2) for mc = 1 (the constant curvature of a sphere of radius 1) is computed in standard spherical coordinates as G = a 0 cos θ + (a 1 sin φ + a 2 cos φ) sin θ.
Equation (3) becomes Therefore, c = 1 m = 0 and, by Theorem 1, the extension of L is 1 2 where we assume without restrictions the constants u 0 , L 0 and W 0 all zero, and where κ ∈ R.
Since |κ| can be multiplied by a positive constant simply by rescaling u, we can assume, if κ = 0, |κ| = m 2 so that the extensions become For κ ∈ C, the explicit form of the extension follows from Remark 2. We remark that the numerical factor m 2 can be absorbed into L by a rescaling of the coordinates (θ, φ). Whenever V depends on (θ, φ) only through trigonometric functions, the rescaling enlights the existence of discrete (polyhedral) symmetries of L on S 2 of order depending on m.
The superintegrable potentials on S 2 with first integrals all quadratic in the momenta are determined in [5], where the sphere is intended, as E 2 previously, as a complex manifold. The nine different superintegrable potentials, up to symmetries in O(3, C) including reflections, are, in Cartesian three-dimensional coordinates (x, y, z) with x = sin θ cos φ, y = sin θ sin φ, z = cos θ [5], where B = a 1 sin φ + a 2 cos φ, C = a 1 cos φ − a 2 sin φ = dB dφ . The second of the subcases of S9 admitting a superintegrable extension is (vi) with G = cos θ. For m = 2 we have for the Hamiltonian H 0 2 the first integral

Extensions of TTW-type systems
Let us consider the Hamiltonian For ζ = 1, χ real and L is a generalization to constant curvature manifolds of the Tremblay-Turbiner-Winternitz system (see [7,8]). We consider the possible extensions of L in dimension three given by Theorem 1. Since the sectional curvature of the metric of L is χ, for mc = χ the general complete solution G of (2) is and the extended Hamiltonian has the form where we assume for simplicity L 0 = u 0 = W 0 = 0. Equation (3) becomes then If a 1 = a 2 = 0, then the equation (14) is satisfied for all F , including the TTW potential with any value of λ. In particular, if λ is rational then L is superintegrable together with its extensions. Otherwise, if a 1 or a 2 are different from zero then the solutions F of the equation (14) can be obtained after observing that ζ (a 1 S ζ (x 2 ) + a 2 C ζ (x 2 )) = − d dx 2 (a 1 C ζ (x 2 ) − a 2 ζS ζ (x 2 )) . Hence, By differentiating the relation a 1 S ζ (x 2 )+a 2 C ζ (x 2 ) = AS ζ (x 2 +ξ), valid for suitable constants A and ξ, we have , a result analogous to the one obtained in [1] for the extension of one-dimensional systems. Indeed, when ζ ∈ N \ {0} then L is in the form of an extension of a one-dimensional system. This is another example of iterative extension.

Conclusions and future directions
We have shown how the procedure of extension proposed in [1] can be used to produce new superintegrable systems starting from the already known ones, together with their first integrals. This procedure allows to extend a number of remarkable systems, including TTW and threeparticle Calogero systems. Moreover, in some cases the procedure can be performed iteratively, thus constructing a family of superintegrable systems in higher dimensions.
Unfortunately not all the superintegrable systems can be extended through our method, but this drawback is balanced by the simplicity and compactness of the algorithm that produces the constants of motion. Further studies are in progress to find a more general form of extension compatible with a larger number of potentials and to analyze iterative extension in other cases. New results about the application to nonconstant curvature manifolds of Theorem 1 have been obtained. The problem of applying the extension procedure to quantum systems is not yet solved: the first integrals described by Theorem 1 cannot be straightforwardly associated with symmetry operators. However, for the quadratic first integrals of type U 2 (G) a first quantization procedure has been considered, quite unsuccessfully, in [2] and then a second one has been studied, leading in suitable cases to symmetry operators. All these progresses will be presented in future publications.