Symmetry, Integrability and Geometry: Methods and Applications Contact Geometry of Hyperbolic Equations of Generic Type

We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge-Ampere (class 6-6), Goursat (class 6-7) and generic (class 7-7) hyperbolic equations, we use Cartan's equivalence method to study the generic case. An intriguing feature of this class of equations is that every generic hyperbolic equation admits at most a nine-dimensional contact symmetry algebra. The nine-dimensional bound is sharp: normal forms for the contact-equivalence classes of these maximally symmetric generic hyperbolic equations are derived and explicit symmetry algebras are presented. Moreover, these maximally symmetric equations are Darboux integrable. An enumeration of several submaximally symmetric (eight and seven-dimensional) generic hyperbolic structures is also given.


Introduction
The purpose of this paper is to give a reasonably self-contained account of some key geometric features of a class of (nonlinear) scalar second order hyperbolic partial differential equations (PDE) in the plane (i.e. in two independent variables) that has received surprisingly very little attention in the literature, namely hyperbolic PDE of generic (also known as class 7-7) type. Even this terminology is not well-known, and so it deserves some clarification.
In the geometric study of differential equations, there is a natural notion of equivalence associated with the pseudo-group of local point transformations, i.e. local diffeomorphisms which mix the independent and dependent variables. Another natural (but coarser) notion is to define equivalence up to the larger pseudo-group of local contact transformations and one of the principal goals of the geometric theory is to find invariants to distinguish different contact-equivalence classes. Restricting now (and for the remainder of this paper) to scalar second order PDE in the plane, we have that given certain nondegeneracy conditions (i.e. one can locally solve the equation for one of the highest derivatives), there is a contact-invariant trichotomy into equations of elliptic, parabolic and hyperbolic type. In the quasi-linear case, invariance of these classes under point transformations appears in [7]. (Inequivalent normal forms are derived in each case.) An elegant geometric proof of invariance under contact transformations in the general case was given by R.B. Gardner [9]. In the hyperbolic case, there exist two characteristic subsystems which give rise to a finer contact-invariant trichotomy into equations of Monge-Ampère (class 6-6), Goursat (class 6-7), and generic (class 7-7) type. While this was known to Vranceanu and almost certainly to E. Cartan and Lie, a modern exposition of these ideas first appeared in [11]. To keep our exposition as self-contained as possible, we include the details of these classifications in this paper. For hyperbolic equations given in the form z xx = f (x, y, z, z x , z y , z xy , z yy ), the (relative) invariants characterizing the three types of hyperbolic equations were calculated parametrically by Vranceanu (c.f. the B, C invariants in [26]). For a general equation F (x, y, z, z x , z y , z xx , z xy , z yy ) = 0, these invariants appeared in Chapter 9 of Juráš' thesis in his characterization of the Monge-Ampère class (c.f. M σ , M τ in [13]). Our derivation of these invariants (labelled I 1 , I 2 in this article) is quite different and the novelty in our exposition (see Theorem 3.3) is obtaining simpler expressions expressed in terms of certain determinants. Moreover, we use these invariants to give general examples of hyperbolic equations of Goursat and generic type (see Table 3).
Hyperbolic equations of Monge-Ampère type have been well-studied in the literature from a geometric point of view (e.g. see [16,17,19,4,5,15,2,14,18] and references therein). This class of equations includes the Monge-Ampère, wave, Liouville, Klein-Gordon and general f -Gordon equations. At the present time and to the best of our knowledge, there exists only one paper in the literature that has been devoted to the study of hyperbolic equations of generic type. This paper, "La géométrisation deséquations aux dérivées partielles du second ordre" [25], was published by Vranceanu in 1937. Despite its appearance over 70 years ago, and much attention having been focused on applications of Cartan's equivalence method in the geometric theory of PDE, very few references to [25] exist. Currently, the paper does not appear on MathSciNet; the only reference to it by internet search engines appears on Zentralblatt Math.
In [25], Vranceanu uses the exterior calculus and Cartan's method of equivalence to study generic hyperbolic equations. One of the most intriguing results of the paper is that all equations of generic type admit at most a nine-dimensional local Lie group of (contact) symmetries. This is in stark contrast to the Monge-Ampère class, where the wave equation is well-known to admit an infinite-dimensional symmetry group. Vranceanu is able to isolate the corresponding maximally symmetric structure equations as well as some submaximally symmetric structures. Furthermore, he is able to integrate these abstract structure equations to obtain an explicit parametrization of the corresponding coframe, leading to normal forms for the contactequivalence classes. As any practitioner of the Cartan equivalence method can probably attest, this is an impressive computational feat. Nevertheless, as in the style of Cartan's writings, Vranceanu's arguments are at times difficult to decipher, hypotheses are not clearly stated or are difficult to discern amidst the quite lengthy calculations, and some of his results are not quite correct. In this paper, we reexamine, clarify, and sharpen some of Vranceanu's results with the perspective of our modern understanding of the geometric theory of differential equations through exterior differential systems and Cartan's equivalence method. The hope is that this exposition will provide a clearer understanding of the geometry of this class of equations for a contemporary audience.
In Section 2 we recall the contact-invariant classification of second order scalar PDE into elliptic, parabolic, and hyperbolic classes based on invariants of a (conformal class of a) symmetric bilinear form, and define the M 1 and M 2 characteristics in the hyperbolic case. This leads to a preliminary set of structure equations for hyperbolic equations. In Section 3, the structure equations are further tightened, and using them we show how the class of M 1 and M 2 leads to the finer classification into equations of Monge-Ampère, Goursat, and generic types. In Theorem 3.3, these subclasses of hyperbolic equations are characterized by means of the relative invariants I 1 , I 2 . We then restrict to the generic case and derive the generic hyperbolic structure equations. We note that in Vranceanu's derivation of the generic hyperbolic structure equations (c.f. (3.2) in this paper), the ǫ = sgn(I 1 I 2 ) = ±1 contact invariant was overlooked. This carries through to the normal forms for the contact-equivalence classes. Section 4 formulates the equivalence problem for generic hyperbolic equations and recalls some facts from Cartan's theory of G-structures applied to our situation. The structure group that we consider here is strictly larger than Vranceanu's, differing by certain discrete components. These naturally arise when considering automorphisms which interchange the M 1 and M 2 characteristics. Both Vranceanu and Gardner-Kamran consider only automorphisms which preserve each of M 1 and M 2 . The nine-dimensional bound on the symmetry group of any generic hyperbolic equation is established in Section 5.
The isomorphism type of the symmetry algebra for the second equation is independent of (ǫ, a) and is non-isomorphic to the symmetry algebra of the first equation. Thus, there are precisely two non-isomorphic (abstract) symmetry algebras that arise for maximally symmetric generic hyperbolic equations. These equations are further distinguished in a contact-invariant way using a contact invariant ∆ 1 and a relative contact invariant ∆ 2 . Both equations satisfy ∆ 1 = 0, but the former satisfies ∆ 2 = 0 while the latter satisfies ∆ 2 = 0.
Let us point out two additional points of discrepancy with Vranceanu's calculations: (1) the restriction of the range of the parameter a to (0, 1], and (2) a missing factor of 2 for the z xy z yy term in the second equation above. The first point is a consequence of the aforementioned larger structure group used in our formulation of the equivalence problem. The additional discrete factors lead to identifications of different parameter values. The second point was minor and the error was only introduced by Vranceanu in the last step of his derivation. To give added justification to the calculation of the normal forms above, we give explicitly the nine-dimensional symmetry algebras for the normal forms listed above. Both equations admit the symmetries The former admits the additional symmetries while the latter admits the additional symmetries The calculation of these symmetries (especially in the latter case) is in general a nontrivial task considering the complexity of the equation. Numerous appendices provide the details of the proofs of the main statements in the body of this article.
All considerations in this paper are local, and we will work in the smooth category. We use the Einstein summation convention throughout. We will make the convention of using braces enclosing 1-forms to denote the corresponding submodule generated by those 1-forms. In general, we will abuse notation and not distinguish between a submodule of 1-forms and its corresponding algebraic ideal (i.e. with respect to the wedge product) in the exterior algebra. This is useful when stating structure equations, e.g. dω 1 ≡ 0 mod I F .
2 Contact-equivalence of PDE Consider a scalar second order PDE F x, y, z, ∂z ∂x , ∂z ∂y , in two independent variables x, y and one dependent variable z. A natural geometrical setting for (2.1) is the space of 2-jets J 2 (R 2 , R) with standard local coordinates (x, y, z, p, q, r, s, t) (Monge coordinates), and the equation above yields a locus We assume that L F is the image of an open subset Σ 7 ⊂ R 7 under a smooth map i F : Σ 7 → J 2 (R 2 , R).
Definition 2.1. We will say that i F is a nondegenerate parametrization of the equation F = 0 if i F has maximal rank and L F is everywhere transverse to the fibers of the natural projection We will always work with nondegenerate parametrizations in this paper. By the transversality assumption (F r , F s , F t ) = 0, and so by the implicit function theorem, one can locally solve (2.1) for one of the highest-order derivatives. Since im((π 2 1 • i F ) * ) = T J 1 (R 2 , R), then (π 2 1 • i F ) * (dx ∧ dy ∧ dz ∧ dp ∧ dq) = 0 and so the standard coordinates (x, y, z, p, q) on J 1 (R 2 , R) along with two additional coordinates u, v may be taken as coordinates on Σ 7 . Thus, without loss of generality, we may assume the parametrization i F has the form i F (x, y, z, p, q, u, v) = (x, y, z, p, q, r, s, t), expressing r, s, t as functions of (x, y, z, p, q, u, v).
The contact system C (2) on J 2 (R 2 , R) is generated by the standard contact forms and pulling back by i F , we obtain a Pfaffian system (i.e. generated by 1-forms) I F on Σ 7 , There is a correspondence between local solutions of (2.1) and local integral manifolds of I F .
Definition 2.2. The equations (2.1) and are contact-equivalent if there exists a local diffeomorphism φ : Σ 7 →Σ 7 such that φ * IF = I F . The collection of all such maps will be denoted Contact(Σ 7 ,Σ 7 ). A contact symmetry is a selfequivalence.
Remark 2.1. More precisely, the above definition refers to internal contact-equivalence. There is another natural notion of equivalence: namely, (2.1) and (2.2) are externally contact-equivalent if there exists a local diffeomorphism ρ : J 2 (R 2 , R) → J 2 (R 2 , R) that restricts to a local diffeomorphismρ : i F (Σ 7 ) → iF (Σ 7 ) and preserves the contact system, i.e. ρ * (C (2) ) = C (2) . It is clear that any external equivalence induces a corresponding internal equivalence, but in general the converse need not hold. The difference between these two natural notions of equivalence is in general quite subtle and has been investigated in detail in [1]. A corollary of their results (c.f. Theorem 18 therein) is that for (2.1), under the maximal rank and transversality conditions, any internal equivalence extends to an external equivalence, and thus the correspondence is one-to-one.
As shown by Gardner [9], the (pointwise) classification of (2.1) into mutually exclusive elliptic, parabolic and hyperbolic classes is in fact a contact-invariant classification which arises through invariants (i.e. rank and index) of a (conformal class of a) symmetric C ∞ (Σ 7 )-bilinear form ·, · 7 on I F , namely where Vol Σ 7 denotes any volume form on Σ 7 . Since i * F is surjective, there exists a 7-form ν on whereφ andψ are any forms on Consequently, letting Vol J 2 (R 2 ,R) = ν ∧ dF , we see that (2.3) is equivalent to where p ∈ Σ 7 . This definition is well-defined: it is independent of the choice ofφ andψ so long as ϕ = i * Fφ and ψ = i * Fψ . A computation in the basis ω 1 , ω 2 , ω 3 reveals that a volume form may be chosen so that . (2.5) Our assumption that i F have maximal rank implies that ·, · 7 cannot have rank zero. Defining we have the following mutually exclusive cases at each point p ∈ Σ 7 : By the commutativity of pullbacks with d, it is clear that this classification is a priori contactinvariant. We remark that in the classical literature on the geometry of hyperbolic equations, the terminology Monge characteristics appears. These are determined by the roots of the characteristic equation The discriminant of this equation (with the coefficients evaluated on F = 0) is precisely − 1 4 ∆, and so the elliptic, parabolic, and hyperbolic cases correspond to the existence of no roots, a double root, and two distinct roots respectively.
In the analysis to follow, all constructions for a PDE F = 0 will implicitly be repeated for a second PDEF = 0 (if present). We will concern ourselves exclusively with the hyperbolic case, that is, an open subset of Σ 7 on which F = 0 is hyperbolic. By the hyperbolicity condition, the two nonzero eigenvalues of ·, · 7 have opposite sign, and hence there exists a pair of rank two maximally isotropic subsystems M 1 and M 2 of I F at every point of consideration.
Remark 2.2. Implicitly, given the Pfaffian system I F corresponding to a hyperbolic PDE F = 0, we assume that a choice of labelling for the M 1 and M 2 characteristics has been made. This is of course not intrinsic. All of our final results will not depend on this choice.
Both Vranceanu [25] and Gardner-Kamran [11] consider only local diffeomorphisms which preserve each of M 1 and M 2 .
The hyperbolicity condition implies that there exists a local basis of I F which by abuse of notation we also denote ω 1 , ω 2 , ω 3 such that and the matrix representing ·, · 7 is in Witt normal form Lemma 2.1 (Preliminary hyperbolic structure equations). There exists a (local) coframe Proof . In Theorem 1.7 in [3], an algebraic normal form for a 2-form Ω is given. In particular, if Ω ∧ Ω = 0, then Ω = σ 1 ∧ σ 2 is decomposable. This statement is also true in a relative sense: if Ω ∧ Ω ≡ 0 mod I, then Ω ≡ σ 1 ∧ σ 2 mod I, where I is an ideal in the exterior algebra. Using this fact, let us deduce consequences of the Witt normal form. By definition of ·, · 7 , we have (taking congruences below modulo I F ) Using ω 1 , ω 2 7 = ω 1 , ω 3 7 = 0, we have and thus dω 1 ≡ 0.
is a (local) coframe on Σ 7 , and the structure equations can be written

Monge-Ampère, Goursat and generic hyperbolic equations
The congruences appearing in the preliminary hyperbolic structure equations can be tightened with a more careful study of integrability conditions and further adaptations of the coframe. The details are provided in Appendix A.
Theorem 3.1 (Hyperbolic structure equations). Given any hyperbolic equation on Σ 7 such that

We have the structure equations
Consequently, we obtain the subclassification of hyperbolic equations given in Table 2.  • Monge-Ampère: wave equation z xy = 0, Liouville equation z xy = e z , Klein-Gordon equation z xy = z, or more generally the f -Gordon equation z xy = f (x, y, z, z x , z y ), and Monge-Ampère equation z xx z yy − (z xy ) 2 = f (x, y).
The terminology for class 6-6 equations is justified by the following result, known to Vranceanu [26]. We refer the reader to Gardner-Kamran [11] for a modern proof. The examples given above were obtained by constructing explicit coframes which realize the abstract structure equations given in Theorem 3.1, which in general is a very tedious task and is equation-specific. We state here two relative invariants I 1 , I 2 (which are related to the two relative invariants U 1 , U 2 ) whose vanishing/nonvanishing determine the type of any hyperbolic equation. Given any hyperbolic equation F = 0, define which are the roots of the characteristic equation (2.6). Without loss of generality, we may assume that F s ≥ 0. (If not, takeF = −F instead.) By the hyperbolicity assumption λ + > 0. The proof of the following theorem is given in Appendix B.
and I i = i * FĨ i . Then we have the following classification of F = 0:

Type Contact-invariant classification
Monge-Ampère I 1 = I 2 = 0 Goursat exactly one of I 1 or I 2 is zero generic Moreover, we have the scaling property: If φ is a function on J 2 (R 2 , R) such that i * F φ > 0, then We note that the scaling property is a fundamental property of these relative invariants: their vanishing/nonvanishing depends only on the equation locus.
Remark 3.1. For a general hyperbolic equation F (x, y, z, p, q, r, s, t) = 0, Juráš' [13] calculated two (relative) invariants M σ , M τ whose vanishing characterizes the Monge-Ampère class. His invariants were given explicitly in terms of two non-proportional real roots (µ, λ) = (m x , m y ) and (µ, λ) = (n x , n y ) of the characteristic equation which he associates to the given PDE. We note here that the characteristic equation (2.6) that we have used differs from that of Juráš (but has the same discriminant). Our invariants I 1 , I 2 appear to be simpler written in this determinantal form.
Using the relative contact invariants I 1 , I 2 we can identify some more general examples of hyperbolic equations of Goursat and generic type.
Corollary 3.2. The classification of hyperbolic equations of the form F (x, y, z, p, q, r, t) = 0, G(x, y, z, p, q, r, s) = 0, and rt = f (x, y, z, p, q, s) is given in Table 3 below.
Proof . The hyperbolicity condition in each case is clear. Define the function and similarly for ∆ G r,s . Without loss of generality G s , f s ≥ 0. The calculation ofĨ 1 ,Ĩ 2 leads to F (x, y, z, p, q, r, t) = 0 :

D. The
G(x, y, z, p, q, r, s) = 0 : For F (x, y, z, p, q, r, t) = 0: Since i * F (F t F r ) < 0, then either I 1 , I 2 both vanish or both do not vanish, i.e. either class 6-6 or class 7-7. The vanishing of I 1 , I 2 is completely characterized by the vanishing of i * F (∆ F r,t ). By Theorem 3.2, we know what all class 6-6 equations of the form F (x, y, z, p, q, r, t) = 0 look like. Hence, i * F (∆ F r,t ) = 0 iff its locus can be given by F (x, y, z, p, q, r, t) = ar + bt + c = 0, where a, b, c are functions of x, y, z, p, q only. The proof for G(x, y, z, p, q, r, s) = 0 is similar and the result for the last equation is immediate.
Remark 3.2. Hyperbolic equations of Goursat and generic type are necessarily non-variational. This is because a variational formulation for a second order PDE requires a first order Lagrangian (density) L(x, y, z, p, q) and the corresponding Euler-Lagrange equation is where D x and D y are total derivative operators Thus, the Euler-Lagrange equation is quasi-linear and, if hyperbolic, is of Monge-Ampère type.
For the remainder of the paper we will deal exclusively with the generic case. In this case U 1 , U 2 in (3.1) are nonzero and can be further normalized through a coframe adaptation. Before carrying out this normalization, we recall some more basic definitions.
Definition 3.2. Given a Pfaffian system I on a manifold Σ, recall that the first derived system I (1) ⊂ I is the Pfaffian system defined by the short exact sequence where π : Ω * (Σ) → Ω * (Σ)/I be the canonical surjection. (Here we abuse notation and identify I with the algebraic ideal in Ω * (Σ) that it generates.) Iteratively, we define the derived flag · · · ⊂ I (k) ⊂ · · · ⊂ I (1) ⊂ I.
We now normalize the coefficients U 1 , U 2 in the generic case. Moreover, explicit generators for the first few systems in the derived flag of C(I F , dM 1 ) and C(I F , dM 2 ) are obtained. The proofs of the following theorem and subsequent corollaries are provided in Appendix C.
3) for some choice of coframe satisfying the above structure equations, we have We will refer to (3.2) and (3.4) collectively as the generic hyperbolic structure equations.

Example 3.2. From
• F (x, y, z, p, q, r, t) = 0 whenever F is not an affine function of r, t.

D. The
Remark 3.4. We have the following dictionary of notations for the adapted coframe labelling: Gardner-Kamran [11] Vranceanu [25] The The structure group and the Cartan equivalence problem In this section, we reformulate the problem of contact-equivalence of PDE as a Cartan equivalence problem. The reader will notice the similarities in the calculation of the structure group in this section and in the calculations in the proof of Corollary 3.4 provided in Appendix C.
Consequently, with respect to the adapted coframe ω on Σ 7 (as specified in Theorem 3.4) and corresponding coframeω onΣ 7 , we have Applying φ * to the dω 1 structure equation in (3.2) yields Similarly, using the dω 2 , dω 3 equations yields Then and so (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 , λ 7 ) = (βa 1 2 , a 1 , βa 1 , βa 1 , β, a 1 , β), This leads us to define and the connected matrix Lie group Let us also define We note that R 4 = S 2 = 1 and we have the relation RS = SR −1 , and consequently R, S generate the dihedral group of order 8 The results (4.1) of the previous calculations establish that where G + is the group generated by G 0 , S, R 2 , which we can realize as the semi-direct product and we have established: The group G will play the role of our initial structure group in the application of the Cartan equivalence method [6,10,22]. Specifically, the Cartan equivalence problem for generic hyperbolic equations can be stated as follows: Given the coframes ω,ω on Σ 7 ,Σ 7 respectively, find necessary and sufficient conditions for the existence of a local diffeomorphism φ : Σ 7 →Σ 7 satisfying (4.3). This is also known as the isomorphism problem for G-structures (ω, G) and (ω, G). [25]) considered the equivalence problem with respect to a smaller group which, in our notation, is G 0 ⋊ R 2 . This has index 4 in G.
The solution of the general Cartan equivalence problem leads to either the structure equations of an {e}-structure or of an infinite Lie pseudo-group. However, for the equivalence problem for generic hyperbolic equations only the former case occurs. In particular, we will show in the next section that we are led to {e}-structures on Σ 7 × G Γ , where G Γ ⊂ G is a subgroup of dimension at most two. (Different {e}-structures arise due to normalizations of nonconstant type, and will depend on choices of normal forms Γ in different orbits.) For the moment, let us recall the general solution to the coframe ({e}-structure) equivalence problem. Our description below is abbreviated from the presentation given in [22].
Let Θ,Θ be local coframes on manifolds M ,M respectively of dimension m, and let Φ satisfy Φ * Θ = Θ. If the structure equations for the {e}-structures are correspondingly then by commutativity of Φ * and d, the structure functions T a bc are invariants, i.e.
For any local function f on M , define the coframe derivatives ∂f ∂Θ a by Let us write the iterated derivatives of the structure functions as where σ = (a, b, c, k 1 , . . . , k s ) and s = order(σ) and 1 ≤ a, b, c, k 1 , . . . , k s ≤ m. We repeat this construction for the barred variables. Necessarily, again as a consequence of commutativity of Φ * and d, the derived structure functions T σ andT σ satisfy the invariance equations Note that these equations are not independent: there are generalized Jacobi identities (which we will not describe explicitly here) which allow the permutation of the coframe derivatives, so in general only nondecreasing coframe derivative indices are needed.
2. The s th order structure map associated to Θ is 3. The coframe Θ is fully regular if T (s) is regular for all s ≥ 0. In this case, let ρ s = rank(T (s) ), and define the rank of Θ as the minimal s such that ρ s = ρ s+1 .

The s th order classifying set is
As a consequence of the invariance equations (4.4), if Θ andΘ are equivalent coframes via Φ : U →Ū , then This is sufficient in the fully regular case. We refer the reader to [22] for a proof of the following theorem.

Nine-dimensional maximal symmetry
The solution to the Cartan equivalence problem (4.3) begins by lifting the problem to the left principal bundles and noting the following key lemma [10].

Lemma 5.1. There exists an equivalence φ as in (4.3) if and only if there exists a local diffeo-
Identifying the coframe ω on Σ 7 with its pullback by the canonical projection Σ 7 × G → Σ 7 , we can writê Using (2.8), the structure equations for these lifted forms are then where the coefficients γ i jk transform tensorially under the G-action and so we can identify the Maurer-Cartan form on G with that on are a basis for the right-invariant 1-forms on G 0 and hence G. Identifying α i on G with their pullback by the canonical projection Σ 7 × G → G, we have the structure equations for the lifted coframe: where η i are semi-basic 1-forms with respect to the projection Σ 7 × G → Σ 7 . The structure equations for the lifted formsω i can be written where a i ρj are constants (c.f. Maurer-Cartan form) andγ i jk is defined as in (5.1).
Definition 5.1. The degree of indeterminacy r (1) of a lifted coframe is the number of free variables in the set of transformations α ρ → α ρ + λ ρ iω i which preserve the structure equations for dω i .
The goal in Cartan's solution algorithm is to reduce to an {e}-structure so that Theorem 4.1 can be invoked. This amounts to essentially adapting the coframes on the base, i.e. fixing a map g : Σ 7 → G. Using Lemma 5.1, coefficients in the structure equations are candidates for normalization, from which the structure group G can be subsequently reduced. However, we only use those coefficients which are not affected by the choice of any map g : Σ 7 → G. Note that pulling the Maurer-Cartan forms back to the base by such a map will express each α ρ in terms of the new coframeω (pulled back to the base). This motivates the following definition.
Definition 5.2. Given a lifted coframe, Lie algebra valued compatible absorption refers to redefining the right-invariant 1-forms α ρ byα ρ = α ρ + λ ρ iω i , where λ ρ i are functions on the bundle. The terms involving the coefficientsγ i jk which cannot be eliminated by means of Lie algebra valued compatible absorption are called torsion terms and the corresponding coefficients are referred to as torsion coefficients.
From (5.2), the dω 5 and dω 7 structure equations indicate thatγ 5 56 ,γ 5 57 ,γ 7 47 ,γ 7 57 are torsion coefficients. Using (3.2), (3.4), and the tensor transformation law (5.1) for the γ's, we see that there is a well-defined G-action on R 4 (i.e. the range of (γ 5 56 , γ 5 57 , γ 7 47 , γ 7 57 )) given by the formulas We can always normalizeγ 5 56 to zero by using the G 0 -action and setting The matrix factorization indicates that we can normalize γ 5 56 to 0 for the base coframe viā This change of coframe is admissible in the sense that it preserves the form of the structure equations in (3.2) and (3.4). (We henceforth drop the bars.) Thus, we have the normal form Γ = (γ 5 56 = 0, γ 5 57 , γ 7 47 , γ 7 57 ). In general, however, this is a normalization of nonconstant type since Γ still may depend on x ∈ Σ 7 . Pointwise, we define the reduced structure group G Γ as the stabilizer of Γ, i.e. it is the subgroup of G preserving the structure equations together with the normalization given by Γ. Clearly, the 1-parameter subgroup generated M (1, 0, a 3 ) (a 3 ∈ R) yields a 1-dimensional orbit through Γ and so dim(G Γ ) ≤ 2 since dim(G) = 3.
The algorithm continues by means of further normalizations and reductions of the structure group until one of two possibilities occurs: 1) the structure group has been reduced to the identity, i.e. get an {e}-structure on Σ 7 , or 2) the structure group has not been reduced to the identity but the structure group acts trivially on the torsion coefficients.
By Theorem 4.1, the former possibility yields a symmetry group of dimension at most seven. In the latter case, the next step in the algorithm is to prolong the problem to the space Σ 7 × G Γ . Here, we have abused notation and written G Γ also for the structure group in the latter possibility above. Since, by Lemma 5.2, r (1) = 0 with respect to the lifted coframe on Σ × G, it is clear that we must have r (1) = 0 for the lifted coframe on Σ 7 × G Γ . Finally, we invoke the following standard theorem (Proposition 12.1 in [22]) written here in our notation:

D. The
In other words, we have prolonged to an {e}-structure on Σ 7 × G Γ . Since dim(G Γ ) ≤ 2 for any choice of Γ, then the symmetry group of the coframe is at most nine-dimensional. Thus, we have proven: Theorem 5.1. The (contact) symmetry group of any generic hyperbolic equation is finite dimensional and has maximal dimension 9.
In fact, this upper bound is sharp. We will give explicit normal forms for all contactequivalence classes of generic hyperbolic equations with 9-dimensional symmetry along with their corresponding symmetry generators and corresponding structure equations. Define and note that although m and n are G 0 -invariant, they are not G-invariant. However, along each G-orbit the product mn is invariant. We define two functions which will play an important role in the classifications to follow. Define Note that ∆ 1 is a contact invariant, and ∆ 2 is a relative contact invariant: it is G + -invariant, but under the R-action,∆ 2 = −ǫ∆ 2 . which yields the normalizationsγ 5 56 =γ 7 47 = 0 and a two dimensional reduction of the initial structure group G. Consequently, the stabilizer G Γ would be at most 1-dimensional and the symmetry group would be at most 8-dimensional. Thus, we must have ∆ 1 = 0.

Complete structure equations
In Appendix D, we provide details of Vranceanu's reduction of the generic hyperbolic structure equations which allowed him to isolate the maximally symmetric and two sets of submaximally symmetric structures.
i=1 on Σ 7 satisfying the generic hyperbolic structure equations (3.2) and (3.4), and the corresponding lifted coframe on Σ 7 × K 0 → Σ 7 . If: 1) all torsion coefficients on which K 0 acts nontrivially are constants, and 2) K 0 cannot be reduced to the identity, then the structure equations can be put in the form and Finally, the integrability conditions for (6.1) (i.e. d 2 ω i = 0 for all i) reduce to the integrability conditions for dm, dn, dB as given above.
Remark 6.1. All structures admitting a 9-dimensional symmetry group are included in (6.1) (since K 0 cannot be reduced to the identity).
If ∆ 1 = 0, then a 2 = a 3 = 0 is the unique solution to (5.6) and G Γ ⊂ K. Alternatively, suppose B = 0. Note thatγ 4 15 andγ 6 17 are torsion coefficients, and for the structure equations (6.1), we have γ 4 15 = γ 6 17 = 0, and the transformation laws (under H 0 ): Consequently, we can normalizeγ 4 15 =γ 6 17 = 0 and reduce the connected component of the structure group to K 0 by setting a 2 = 0. The discrete part of the structure group will preserve this reduction since Let us now examine in detail the case when m, n are constants. Then (6.2) becomes Applying d to (6.4) and simplifying, we obtain the integrability condition

All of the above structures have a symmetry group with dimension at least seven.
Proof . We prove only the final assertion as the others are straightforward to prove. Let G Γ be the reduced structure group for which there is no group dependent torsion. By construction (c.f. Theorem 6.1), we must have K 0 ⊂ G Γ , and by Proposition 5.1 we prolong to an {e}structure on Σ 7 × G Γ . If B is constant, then by Theorem 4.1 the symmetry group has dimension dim(Σ 7 × G Γ ) ≥ 8. If B is nonconstant, then by Corollary 6.1, G Γ ⊂ K. Note thatB = B, so equation (6.4) implies that on Σ 7 × G Γ , we have dB = −2 (2ǫm∆ 1 + nB)ω 5 + 2 (2n∆ 1 + mB)ω 7 . The results are organized in Table 4 according to the dimension of the symmetry group of the resulting {e}-structures on Σ 7 × G Γ .

D. The
For ease of reference, we state below the structure equations explicitly for each of the cases above. For the maximally symmetric cases, we state the structure equations for both the base coframe {ω 1 , . . . , ω 7 } and the lifted coframe on Σ 7 × G Γ . In the submaximally symmetric cases, we only display structure equations for the lifted coframe. (One can obtain the structure equations on the base simply by settingα 1 = 0 and removing all hats from the remaining variables.) In each case, we assume that G Γ and all parameters are as in Table 4. Note that dω i are determined by (5.3). Following potentially some Lie algebra valued compatible absorption, α ρ = α ρ + λ ρ iω i , the structure equations dα ρ are determined by the integrability conditions d 2ωi = 0. (We only display the final results.) For those coframes whose structure equations depend explicitly on the (nonconstant) function h, we have m = 0 (c.f. Table 4) and the symmetry algebra is determined by restricting to the level set h = h 0 , where h 0 is a constant. (Note: We will abuse notation and identify h ∈ C ∞ (Σ 7 ) with its pullback to the bundle.) On this level set, we have 0 = dh = −nω 5 + mω 7 . We can choose (the pullback of) {ω 1 , . . . ,ω 6 ,α 1 } as a coframe on each level set, and the corresponding structure equations will have constant coefficients. Thus, these are Maurer-Cartan equations for a local Lie group. A well-known fact is that the isomorphism type of the symmetry algebra of a coframe determined in this way is independent of the level set chosen. Consequently, we make the canonical choice and restrict to the level set h = 0 in these cases.
The structure constants for the (contact) symmetry algebra for each of the structures can be read off from the structure equations for the coframe (or its pullback to the level set h = 0 if h appears explicitly). Only the symmetry algebras appearing in the 9-dimensional case will be studied in further detail in this article.

Normal forms
Let us determine how the coordinates (u, v) on Σ 7 are related to the standard 2-jet coordinates (x, y, z, p, q, r, s, t) ∈ J 2 (R 2 , R) . Let χ : R 2 → Σ 7 be any integral manifold of I F with independence condition χ * (dx ∧ dy) = 0. Without loss of generality, we identify the coordinates (x, y) on R 2 with the (x, y) coordinates on Σ 7 . The composition i F • χ is then an integral manifold of the contact system C (2) and on R 2 we can write dp = rdx + sdy, dq = sdx + tdy, where for convenience p is identified with (i F • χ) * p, and similarly for the coordinates q, r, s, t. Substituting (7.2) into the conditions 0 = χ * ω 2 = χ * ω 3 , and extracting the coefficients of dx and dy, we obtain the relations 0 = 6vs + 6r + 2ǫmu 3 − 3ǫm 2 u 2 v − αv 3 ,

Nine-dimensional symmetry algebras
The calculations leading to (7.3) are quite long and consequently to confirm the validity of (7.3) (and in turn, Theorem 7.1), it is useful to describe the nine-dimensional (contact) symmetry algebra explicitly for the normal forms given in the previous section. Calculating the symmetry algebra is a nontrivial task however -the standard Lie method of calculating symmetries (by working in J 2 (R 2 , R) on the equation locus) is highly impractical owing to the complexity of the equations. In Appendix F, we describe how the symmetry algebra was found by an alternative method. In order to give a unified description of the symmetry algebras, we work with the normal forms (7.4) and (7.5) as these arise from the parametrization (7.3).
Proposition 7.1. Any equation of the form F (r, s, t) = 0 admits the symmetries The equations (7.4) and (7.5) have the following additional symmetries: In particular, all of these symmetries are projectable point symmetries.
(Recall that a point symmetry here is a vector field on J 0 (R 2 , R). A point symmetry is projectable if it projects to a vector field on the base R 2 .) The normalization of (7.5) to (7.6) is carried out by lettingx = 1 α x from which we get: Corollary 7.1. The generic hyperbolic equation (7.6) has symmetry generators X 1 , . . . , X 6 as in Proposition 7.1 as well as We will denote the corresponding abstract Lie algebras as g α and express their commutator relations in a canonical basis. Let (e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 , e 8 , e 9 ) = (X 2 , X 3 , X 4 , X 5 , −X 8 , X 7 , X 1 , −2X 6 + X 7 , −X 9 ).
Theorem 7.2. The contact symmetry algebra of any maximally symmetric generic hyperbolic PDE is: 2) contact-equivalent to a (projectable) point symmetry algebra.
Moreover, there are exactly two isomorphism classes of Lie algebras (represented by g 0 and g 1 ) that arise as such symmetry algebras.
The radicals of g 0 and g 1 can be identified in the latter classification as

External symmetries
Given any vector field X on J 0 (R 2 , R), there is a corresponding prolonged vector field X (2) on J 2 (R 2 , R). This prolongation is uniquely determined by the condition that L X (2) C (2) ⊂ C (2) , where C (2) is the contact system on J 2 (R 2 , R). See (F.5) for the standard prolongation formula. For the vector fields in Proposition 7.1, we have For (7.4) or (7.5), we verify the external symmetry condition Clearly this is satisfied by X (2) i , i = 1, . . . , 6 since they have no components in the ∂ ∂r , ∂ ∂s , ∂ ∂t direction and since F = F (r, s, t) for (7.4) and (7.5). For the remaining vector fields we have (7.4) : and so the external symmetry condition is satisfied.

Internal symmetries
The symmetry generators X (2) i are all tangent to the equation manifold F = 0, so they induce (via the parametrization (7.3)) corresponding vector fields Z i on Σ 7 . Letting X denote the projection onto J 1 (R 2 , R), and identifying the coordinates (x, y, z, p, q) on J 1 (R 2 , R) with corresponding coordinates on Σ 7 , we have with u = w + mv. One can verify directly that these vector fields satisfy the internal symmetry condition where I F = ω 1 , ω 2 , ω 3 is given by the explicit coframing (7.1).

Darboux integrability
Definition 7.1. For a hyperbolic PDE F = 0, I F is said to be Darboux-integrable (at level two) if each of C(I F , dM 1 ) and C(I F , dM 2 ) contains a completely integrable subsystem of rank two that is independent from I F .
Recall that for our adapted coframe as in Theorem 3.4, we have Theorem 7.3. Given a generic hyperbolic PDE F = 0 with (maximal) 9-dimensional symmetry group, the second derived systems C(I F , dM 1 ) (2) and C(I F , dM 2 ) (2) :

1) are completely integrable, and hence I F is Darboux integrable, and
2) contain rank one completely integrable subsystems.

D. The
Abstractly, Darboux's integration method for these systems proceeds as follows. Darboux integrability of I F implies the existence of completely integrable subsystems J i ⊂ C(I F , dM i ). Applying the Frobenius theorem to each subsystem J i , there exist local functions f i , g i called Riemann invariants such that If ϕ 1 , ϕ 2 are arbitrary functions, then restricting to any submanifold determined by the structure equations (2.7) become whereĨ F = {ω 1 ,ω 2 ,ω 3 } is the restriction of I F to S. Hence,Ĩ F is completely integrable, and so there exist local functions h 1 , h 2 , h 3 on S such that Hence, these functions h 1 , h 2 , h 3 are first integrals ofĨ F , and together with the constraint S determine first integrals of I F . Explicitly, from our parametrization of the coframe {ω i } 7 i=1 on Σ 7 (c.f. (7.1)), we have: and so  [8], this statement is incorrect and the equation 3rt 3 + 1 = 0 is a counterexample. Moreover, as described above, all maximally symmetric generic hyperbolic equations have Riemann invariants. We refer the reader to page 130 in Goursat [12] for the implementation of Darboux's method to the equation 3rt 3 +1 = 0. The implementation of Darboux's method in the case (ǫ, a) = (1, 1) appears to be computationally quite difficult.
Let us comment on Darboux integrability for the submaximally symmetric cases described in Table 4. Recall that the structure equations listed in Sections 6.2 and 6.3 are those for the lifted coframe. To obtain the structure equations for the corresponding base coframe, we simply setα 1 = 0 and remove all hats. For all these cases we have either that and hence I F is Darboux integrable, or and I F is not Darboux integrable. We list the possibilities in Table 5. Moreover, for these submaximally symmetric cases, all which are Darboux integrable have one-dimensional subsystems of C(I F , dM 1 ) (2) and C(I F , dM 2 ) (2) which are completely integrable (namely, {ω 5 } and {ω 7 } respectively). Thus, the converse of Theorem 7.3 is clearly false.

Concluding remarks
Let us summarize some of the main results of this paper: • We derived relative invariants I 1 , I 2 (see Theorem 3.3) given parametrically in terms of an arbitrary hyperbolic equation F (x, y, z, z x , z y , z xx , z xy , z yy ) = 0. Their vanishing/nonvanishing distinguishes the three types of hyperbolic equations.
• In the abstract analysis of the generic hyperbolic structure equations, we identified relative contact invariants m, n, B and ∆ 1 = mn + ǫ, ∆ 2 = m 2 − ǫn 2 which played a key role in the classification of various generic hyperbolic structures admitting nine, eight, and seven-dimensional symmetry along with the corresponding complete structure equations.
• Integration of maximally symmetric structure equations, leading to normal forms for all contact-equivalence classes of maximally symmetric generic hyperbolic equations.
• Nine-dimensional symmetry algebras for these normal forms for generic hyperbolic equations are given explicitly. There are exactly two such nonisomorphic algebras.
• For any maximally symmetric generic hyperbolic equation, the second derived systems of C(I F , dM i ), i = 1, 2 are rank 2 and completely integrable. Hence, all maximally symmetric generic hyperbolic equations are Darboux integrable.
We conclude with some possible points for future investigation: 1. Maximally symmetric equations: (1) Do "simpler" normal forms exist? (2) Implement Darboux's integration method in the general case. (3) Investigate the existence of conservation laws. (4) Study the local solvability of these equations.
2. Submaximally symmetric equations: Integrate the structure equations given in Sections 6.2 and 6.3 and find normal forms for the corresponding PDE equivalence classes. Address similar questions as above.
3. The submaximally symmetric structures that we have derived here (see Table 4 and Sections 6.2 and 6.3) share the common property that m, n are constants and K 0 is a subgroup of the structure group. Are there any other reductions of the initial 3-dimensional structure group that lead to valid structures? 4. In this article, we have carried out a detailed analysis of the generic (7-7) case. Hyperbolic equations of Goursat (6-7) type are equally poorly understood. Some preliminary results on structure equations were stated in [11], but to our knowledge, Vranceanu's student Petrescu [23] has written the only paper which has made a more detailed investigation into the contact geometry of the Goursat class. Recasting Petrescu's results for a contemporary audience and building upon his work would make for a natural sequel to our paper.

A Hyperbolic structure equations
We give here the details of the proof of Theorem 3.1 starting from the preliminary hyperbolic structure equations (2.7). The main details of this proof have appeared in [25] and [11].
Note A.1. In this section, we will define changes in the coframe basis using a "bar", e.g. ω i = g i j ω j , but we make the convention that the bar is immediately dropped afterwards, i.e. so that ω i is redefined to beω i .
This preserves both the dω 2 and dω 3 structure equations (sinceω 6 ∧ω 7 = ω 6 ∧ ω 7 ). We have , we may without loss of generality take b 4 = b 5 = b 6 = 0. (Note that the dω 3 equation is not affected.) Setting b 7 = U 1 , we have A similar argument modulo {ω 1 , ω 3 } and for the M 2 characteristic system yields and we emphasize that ω i in (A.2) are the same as ω i in (A.3). This implies We calculate The first equation implies that γ 4 67 = 0 (c.f. as defined in (2.8)) and hence the second equation implies W = 0.

B Principal contact invariants for hyperbolic equations
Here we derive the contact invariants stated in Theorem 3.3. We give a brief outline of the computations to follow. Ultimately, we are looking for a coframe satisfying the hyperbolic structure equations given in (3.1). Beginning with the pullback of the basis θ 1 , θ 2 , θ 3 of the contact ideal on J 2 (R 2 , R), we find a canonical basisθ 1 ,θ 2 ,θ 3 whose pullback brings ·, · 7 into the Witt normal form. This yields the basisω 1 , . . . ,ω 7 in Lemma B.2. A subsequent normalization in (B.5) defines the basis ω 1 , . . . , ω 7 which satisfies (3.1) and from which (multiples of) U 1 and U 2 can be extracted in parametrized form. The final calculation of these parametrized functions is performed in the ambient space J 2 (R 2 , R), essentially using the simple fact given in (2.4). Pulling these back by i F yields the desired invariants. We begin with the following lemma.
Lemma B.1. Given a symmetric bilinear form represented by the change of basis matrix brings Q, up to a nonzero scaling, into the Witt normal form, i.e.
We apply this elementary result to the 2 × 2 submatrix appearing in the matrix in (2.5) representing the bilinear form ·, · 7 , with respect to the basis θ 2 , θ 3 , and arrive at the canonical basis Without loss of generality, we assume that F s ≥ 0. (If not, consider the equation −F = 0 which defines the same locus as F = 0.) In the notation of the previous lemma, we have Note that λ + > 0, but λ − may vanish. Both λ ± are roots of the polynomial Furthermore, let us note the identities We also define the total derivative operators is a (local) coframe on Σ 7 satisfying the structure equations where Proof . We need to show thatω is linearly independent, or equivalentlyω 1 ∧ · · · ∧ω 7 = 0. By (2.4), it suffices to show thatθ 1 ∧ · · · ∧θ 7 ∧ dF = 0. We calculatẽ θ 1 ∧θ 2 ∧θ 3 ∧θ 4 ∧θ 6 = 4|∆|dz ∧ dp ∧ dq ∧ dx ∧ dy (B.3) and soθ Thus,ω is a coframe on Σ 7 . Let us verify the structure equations. Note that where all coefficients on the right side are pulled back by i * F to Σ 7 (i.e. evaluated on F = 0) Mod M 2 , we have Now examine dω 2 and dω 3 . Note the relation and so, Mod M 1 , we have Let us write We make the change of basis , to obtain the structure equations . Thus, in the notation of Theorem 3.1, we have that U 1 = b 7 and U 2 = c 5 and by the argument given at the end of Appendix A, we must have that W = 0.
We are primarily concerned with the vanishing / nonvanishing of U 1 and U 2 , though the sign will play a later role in the determination of the ǫ contact invariant. It suffices to calculate Note that the sign of these coefficients is reversed for both since ρ < 0. As a further simplification, since i * F (dx), i * F (dy) depend only onω 4 ,ω 6 , and since i * F (λ + ) > 0, it suffices to calculate Let us illustrate how to calculate theω 7 . Thus, we calculatef 7 using θ 1 ∧ · · · ∧θ 6 ∧ d F t λ + ∧ dF =f 7θ 1 ∧ · · · ∧θ 7 ∧ dF and then use i * F to obtain f 7 . Using (B.3), we have Let I 1 = i * FĨ 1 , and note I 1 and f 7 = i * Ff 7 have opposite sign. Because of the aforementioned sign reversal (i.e. U 1 and f 7 have opposite sign), we have sgn(U 1 ) = sgn(I 1 ). (B.7) We perform a similar computation for the ω 5 coefficient in i * F d Fr λ + . For the scaling property, suppose thatF = φF with i * F φ > 0. We may without loss of generality suppose thatΣ 7 = Σ 7 and iF = i F . Hence, Since i * F commutes with d, From the coframe definition in Lemma B.2, we see that Hence, from the expression for ν 1 in (B.6), Since i * F∆ = i * F (φ 2 ∆), then by (B.6) and (B.9) we see that necessarilŷ Similarly, we find that C Generic hyperbolic structure equations Starting with the hyperbolic structure equations (c.f. Theorem 3.1), we specialize here to the generic case and prove Theorem 3.4 and Corollaries 3.3 and 3.4.
Similarly, we describe the derived flag of C(I F , dM 2 ).
and so Consequently, we must have δ = ǫ and so ǫ is a contact invariant. Finally, from (B.7), (B.8), and the proof of Theorem 3.4 given in Appendix C, we have ǫ = sgn(U 1 U 2 ) = sgn(I 1 I 2 ).

D Isolating maximally and submaximally symmetric structures
Starting with the structure equations (3.2), (3.4) (and not assuming γ 5 56 = 0) we clarify Vranceanu's method of isolating all structures admitting maximal symmetry as well as several structures admitting submaximal symmetry. The key to understanding Vranceanu's method occurs in a single rather cryptic paragraph on pages 367-368 of [25]. In our notation, Vranceanu writes: In effect, we can remark that if a 1 were equal to 1 we could reduce a 2 , and consequently a 3 , to zero, by cancelling in the covariant dω 6 the term γ 6 46 with the help of the coefficient a 2 . This indicates that for a 2 , a 3 different than zero, we must have a 1 = 1, and consequently the systems for which we cannot reduce a 2 and a 3 to zero are found amongst those for which a 1 = 1, a 2 = a 3 = 0, and such that we cannot reduce a 1 to 1.
Let us refer back to the structure equations (5.2) for the lifted coframe on Σ 7 × G → Σ 7 . Let H 0 denote the subgroup obtained by setting a 1 = 1 in G 0 . Since Vranceanu did not consider discrete symmetries, let us consider the corresponding lifted coframe on Σ 7 ×H 0 → Σ 7 . We would have structure equations as in (5.2) except no α 1 terms would appear (and dα 2 = dα 3 = 0). Thus, using Lie algebra valued compatible absorption using α 2 and α 3 only, γ 6 46 would be a torsion coefficient. With respect to the group H 0 , we have the transformation lawŝ γ 6 46 = γ 6 46 − γ 6 56 a 3 + a 2 ,γ 6 56 = γ 6 56 . (D.1) By setting a 2 = γ 6 56 a 3 − γ 6 46 , we can normalizeγ 6 46 = 0. In the group reduced from H 0 , we only obtain a 2 = γ 6 56 a 3 , so it is unclear whether a 2 , a 3 can both be reduced to 0. Thus, Vranceanu's claim may be true, but if so it is certainly not obvious. Are there any structures for which a 1 can be reduced to 1, but at least one of a 2 , a 3 are nonzero?
The second sentence in the paragraph appears to be quite cryptic and at first sight appears even self-contradictory. One can make sense of this as follows: Vranceanu sets out to find all structures for which K 0 = {diag(a 1 dω 7 = η 71 ∧ω 1 + η 73 ∧ω 3 + η 76 ∧ω 6 + η 77 ∧ω 7 , where η i are semibasic with respect to the canonical projection Σ 7 × K 0 → Σ 7 . Note that under the K 0 -action, all coefficients transform by a scaling action: for some scaling weight p ∈ R. (In other words, by setting a 2 = a 3 = 0 in the general transformation formulas under G 0 , c.f. last sentence in Vranceanu's paragraph above.) For specific i, j, k, if γ i jk = 0, then we can normalize toγ i jk = ±1 by setting a 1 = |γ i jk | −1/p , thereby reducing K 0 to the identity. Since we are assuming that K 0 cannot be reduced, Vranceanu's conclusion would be that γ i jk = 0. A technical assumption that should be made here is that if p = 0, we are considering γ i jk to be a constant torsion coefficient. For example, if γ i jk = 0 (as functions) but vanishes at a point, then K cannot be normalized since a 1 must be nonzero. In summary, we have: Lemma D.1. Supposeκ = (a 1 ) p κ is a torsion coefficient with respect to K 0 . If (1) K 0 cannot be reduced to the identity, (2) p = 0, and (3) κ is a constant, then κ = 0.
Note D.1. We do not need to assume that all torsion terms are constant. Only those torsion terms with a nontrivial scaling action by K 0 are assumed to be constant.
Thus, the structure equations take the form where η k = γ k kℓ ω ℓ (no sum on k), and Proof . Let us evaluate some integrability conditions.
We use these expressions for the coframe derivatives to rewrite the system (E.6), (E.7) and obtain expressions for σ, µ, ρ. The three equations in (E.6) are respectively equivalent to These three vector fields are also tangent to the equation manifold, but they are clearly not contact vector fields since: (1) they do not preserve the contact ideal, and (2) they do not arise as prolongations of vector fields on J 1 (R 2 , R) (c.f. Bäcklund's theorem). However, this leads us to the following problem: Do (F.2)-(F.4) describe the r, s, t components of contact vector fields on J 2 (R 2 , R)? If so, then those contact vector fields would also be tangent to the equation manifold and hence would correspond to contact symmetries of the equation. The search is greatly simplified by the observation that the components in (F.2)-(F.4) are linear in x, y, r, s, t and independent of z, p, q. By Bäcklund's theorem, any contact vector field on J 2 (R 2 , R) is the prolongation of a contact vector field on J 1 (R 2 , R). Hence, we look at a generalized vector field of order one on J 0 (R 2 , R) X = ξ 1 (x, y, z, p, q) ∂ ∂x + ξ 2 (x, y, z, p, q) ∂ ∂y + η(x, y, z, p, q) ∂ ∂z and examine its prolongation X (2) = pr (2) (X) on J 2 (R 2 , R). If we write then the standard prolongation formula [22] is where D i are total derivative operators and we have used the notation x 1 = x, x 2 = y. J is an unordered (symmetric) multi-index, so that for example z 1 = p, z 12 = z 21 = s, etc. Ian Anderson's JetCalculus package in Maple v.11 was very useful for computing and manipulating these prolongations. We give an outline of the calculation here.