Meta-Symplectic Geometry of $3^{\rm rd}$ Order Monge-Amp\`ere Equations and their Characteristics

This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order (2D) Monge-Amp\`ere equations, by using the so-called"meta-symplectic structure"associated with the 8D prolongation $M^{(1)}$ of a 5D contact manifold $M$. We write down a geometric definition of a third-order Monge-Amp\`ere equation in terms of a (class of) differential two-form on $M^{(1)}$. In particular, the equations corresponding to decomposable forms admit a simple description in terms of certain three-dimensional distributions, which are made from the characteristics of the original equations. We conclude the paper with a study of the intermediate integrals of these special Monge-Amp\`ere equations, herewith called of Goursat type.


Introduction
Classical Monge-Ampère equations (MAEs with one unknown function and two independent variables) constitute a distinguished class of scalar 2 nd order (nonlinear) PDEs owing to two remarkable properties: first, they can be, in a sense, reconstructed from their characteristics [2] and, second, their so-called "characteristic cone" 1 degenerates into the union of zero, one or two 2D planes (according to the elliptic, parabolic or hyperbolic character of the equation). By contrast, the characteristic cone of multidimensional MAEs, which are defined by a linear combination of the minors of an Hessian matrix equated to zero, is the union of n-dimensional linear subspaces only in special cases, dubbed "Goursat-type" [2,13]. Thus, in the classical, i.e., two-dimensional, case the general MAEs and the MAEs of Goursat type form a single class.
The primary motivation of this paper was to determine whether, and to what extent, such a phenomenon occurs also in the context of 3 rd order PDEs. Indeed, the above notion of "characteristic cone" is not exclusive to 2 nd order PDEs (though it is far more exploited in this context), pertaining to PDEs of arbitrary order and number of (in)dependent variables. This very introduction is the appropriate place to acquaint the reader with such a notion, which it is going to be the main gadget for our analysis. Putting off an intrinsic geometric definition (see (9) and (78) later on), we deemed it convenient to introduce it here in a friendly coordinate way.
A curve (4) is characteristic for E at the point 3 m k−1 ∈ E, i.e., a point (5) m k−1 = (x 1 , x 2 , u, . . . , p i1···i l , . . . ), l ≤ k, whose coordinates satisfy (1), if there exists a tangent vector ν =Φ(t 0 ) at the point (6) m k−2 = Φ(t 0 ) = (x 1 , x 2 , u, . . . , p i1···i l , . . . ), l ≤ k − 1, such that 1 This definition is used here for the first time, though the notion behind it is rather old, and may have appeared in other guises someplace else. For instance, it is called "fold-type singularity equation" in [27]. 2 We adopted the same notation used in [28], where the reader may also find a gentle introduction to the theory of characteristics and singularities of nonlinear PDEs. 3 The choice of notation "m k−2 " and "m k−1 " will be motivated later on. Figure 1. If coordinates (x 1 , x 2 , u, . . . , p i1···i l , . . . ), with l ≤ k − 1, corresponding to the red linear space, are held fixed, then a PDE E reduces an hypersurface in the (. . . , p i1···i l , . . .)-space, with l = k, depicted as a grey smooth object inside a blue linear container. A tangent direction ν to the red space can be coupled with the differential of F at a point m k−1 ∈ E via (7): those yielding zero form the cone V E m k−2 , the number and multiplicity of whose sheets are determined by the degree k polynomial appearing at the left-hand side of (7).
From equation (7) it is clear that one can associate with any point (5) of E a number ≤ k of directions (8) in the space with coordinates (x 1 , x 2 , u, . . . , p i1···i l , . . . ), l ≤ k − 1. So, if we keep the point (6) fixed and let the point (5) vary in E, the aforementioned directions form, in general, a number ≤ k of cones (polynomial (7) might possess multiple and/or imaginary roots). We call such set of directions (see Fig. 1) the characteristic cone of E at m k−2 and denote it by (9) V E m k−2 := {Directions ν as in (8) | ∃m k−1 ∈ E such that (ν 1 , ν 2 ) satisfies (7)}. Intuitive definition of the characteristic cone of E at m k−2 .
In this paper we deal with the interesting issue of studying a PDE E through the above defined characteristic cone V E : such a matter has been given a relatively marginal attention and its potential is not yet fully understood. It should be stressed, however, that E cannot be completely replaced by V E in all contexts, having a profoundly different nature. Nevertheless, the study of V E can be very convenient, for instance, in facing classification problems. For example, the only (bidimensional) 2 nd order PDEs E whose V E degenerates into two 2D planes are the classical MAEs (see, e.g., [4,3,21]), so that the problem of classifying (classical) MAEs entails classifying 2D certain distributions on a 5D contact manifold [4]. A similar result holds true in the multidimensional case, but only for the class of MAEs introduced by Goursat [13].
The step ahead we are going to take here is the following. Basically, we study those PDEs that correspond to 3D distributions on the 8D first prolongation of a 5D contact manifold, borrowing the key ideas from the above works. In spite on the evident methodological analogy with the 2 nd order case, the 3 rd order one will display a few unexpected peculiarities, mostly due to the richer structure of prolonged contact manifolds. It should be underlined that, as a byproduct of this study, we obtain an invariant definition of 3 rd order bidimensional MAEs which, unlike 2 nd order MAEs, intensively studied since the second half of XIX century by many mathematicians such as Darboux, Lie, Goursat and, more recently, by Lychagin [19], Morimoto [24] and their school in the context of contact and symplectic geometry, have been defined just recently by Boillat [7] as the only PDEs which satisfies the property of complete exceptionality in the sense of Lax [20]. It seems that a systematic analysis of such PDEs in a differential-geometric context is still lacking as, up to now, no serious effort has been made to extend the classical theory to the case of 3 rd order MAEs, is spite of their importance [1,10,12,26]. Much as in the geometric approach to classical MAEs it is conventient to exploit the contact/symplectic geometry underlying 2 nd order PDEs, to deal with 3 rd order MAEs we shall make use of the prolongation of a contact manifold, equipped with its Levi form (see, e.g., [25], Section 2), a structure known as meta-symplectic, quickly reviewed below.
Classical MAEs ↔ contact manifolds and symplectic geometry 3 rd order MAEs ↔ prolongation of contact manifolds and meta-symplectic geometry 1.2. Structure of the paper. The exposition is pivoted around the central Section 6, where we prove the main results, Theorems 2, 3 and 4, whose statements are given earlier in Section 3. As their formulation requires simple but nonstandard generalizations of some basic tools of contact/symplectic geometry, we added the preliminary Section 2 with the idea of introducing the three theorems above in the most self-contained way. Moreover, between their statement and the proof, we slotted two more sections to accommodate some preparatory results. More precisely, in Section 4, after reviewing the standard notions of rank-one vectors and characteristics covectors, we introduce the key notion of "threefold orthogonality" and study its main properties, while in the subsequent Section 5 we go deeper into the analysis of the main gadget of our study, the above defined characteristic cone (9), with a particular emphasis on its relationship with the more widely known characteristics variety. In Section 7 we study the intermediate integrals of Goursat-type 3 rd order MAEs in terms of their characteristics, thus providing a tangible application of the main results. Their full applicative range, however, goes beyond the scope of this paper, and we could only give it a quick try in the concluding Section 8.

Conventions.
We make an extensive usage of the symbol "P" in order to avoid too many repetitions of the sentence "up to a conformal factor". Nevertheless, if this latter is born in mind, all projective bundles can be replaced by their linear counterparts (taking care of increasing all dimensions by 1). One-dimensional linear object may be identified with their generators. As a rule, if P → X is a bundle, and x ∈ X, we denote by P x the fiber of it at x. Symbol D denotes the derived distribution [D, D] of a distribution D. If there is any vertical distribution in the surrounding manifold, D v denotes the intersection of a distribution D with the vertical one. Differential forms on a manifold N (resp., distribution D) are denoted by Λ * N (resp., Λ * D * ) and f * is the tangent map of f : N → Q. If T is a tensor on N , or a distribution, we sometimes skip the index "n" in T n , if it is clear from the context that T has been evaluated in n ∈ N . As a rule, we use the term "line" for contravariant quantities, and "direction" for covariant ones, and sometimes we gloss over the distinction between "the symbol of E = {F = 0}" and "the symbol of E". The symmetric tensor product is denoted by . The Einstein summation convention will be used, unless otherwise specified. Also, when a pair jk (resp., a triple ijk) runs in a summation, such summation is performed over 1 ≤ j ≤ k ≤ 2 (resp., 1 ≤ i ≤ j ≤ k ≤ 2), unless stated differently. See also Table 1. Throughout the paper, by M (0) , C 0 , and m (0) we shall always mean M , C, and m, respectively.

2.
Preliminaries on (prolongations of) contact manifolds, and (meta)symplectic structures 2.1. Contact manifolds, their prolongations and PDEs. We introduce here a minimal set of notions needed to lay down an invariant coordinate-free definition of the characteristic cone, superseding (9). Throughout this paper, (M, C) will be a 5D contact manifold. In particular, C is a completely non-integrable distribution of hyperplanes (i.e., 4D tangent subspaces) on M locally described as C = ker θ, where the 1-form θ is determined up to a conformal factor and θ ∧ dθ ∧ dθ = 0. The restriction dθ| C defines on each hyperplane C m , m ∈ M , a conformal symplectic structure: Lagrangian (M, C) a 5D contact manifold θ the contact form on M L(C m ) the Lagrangian Grassmannian of C m M (1) π −→ M the Lagrangian bundle Ω the meta-symplectic form on M (1) (i.e., maximally dθ-isotropic) planes of C m are tangent to maximal integral submanifolds of C and, as such, their dimension is 2. We denote by L(C m ) the Grassmannian of Lagrangian planes of C m and by (10) π : the bundle of Lagrangian planes, also known as the 1 st prolongation of M . The key property of the manifold M (1) is that it is naturally endowed with a 5D distribution, defined by (1) considered as a Lagrangian plane in C m . Let us denote by θ (1) the distribution of 1-forms on M (1) vanishing on C 1 . Then, by definition, a Lagrangian plane of M (1) is a 2D subspace which is π-horizontal and such that both distributions of forms θ (1) and dθ (1) vanish on it. In analogy with (10), we define the 2 nd prolongation M (2) of a contact manifold (M, C) as the first prolongation of M (1) , that is Observe that, even if the operation of prolongation (10) is formally identical to (12), it lacks the requirement of horizontality, due to the absence in M of a fibred structure. Nevertheless, projection (10) is the beginning of a tower of natural bundles which, for the present purposes, will be exploited only up to its 2 nd term. It is well known that π 2,1 is an affine bundle (see, e.g., [16]).
On the other hand, if horizontality is dropped in (12), one augments M (2) with the so-called "singular" integral elements of C 1 (which, in the case of PDEs, formalize the notion of "singular solutions": see, e.g., [22,27,28]) thus obtaining a projective bundle M (2) , which can be thought of as the "compactification" of M (2) . Such a compactification is defined by the same formula (12), but without the horizontality condition on Lagrangian planes.
Previous constructions lead naturally to consider a geometrical object which will play a prominent role in our analysis, namely the so-called tautological bundle over M (i) (14) L where the fiber L m i is m i itself, understood as a 2D subspace of C i−1 , with i = 1, 2. We keep the same symbol L for both the tautological bundles over M (1) and M (2) , since it will be always clear from the context which is which. Now we focus on the local description of the geometrical objects just introduced. There exist coordinates (x i , u, p i ) on M , i = 1, 2 such that θ = du − p i dx i , called contact coordinates, such that C is described by where D i is the total derivative with respect to x i , truncated to the 0 th order. A system of contact coordinates (x i , u, p i ) induces coordinates on M (1) as follows: a point m 1 ≡ L m 1 ∈ M (1) has coordinates (16) iff m = π(m 1 ) = (x i , u, p i ) and the corresponding Lagrangian plane L m 1 is given by: Similar reasonings lead to the following local descriptions where now D i stand for total derivatives truncated to the 1 st order and a point m 2 ∈ M (2) has coordinates (x i , u, p i , p ij = p ji , p ijk = p ikj = p jik = p jki = p kij = p kji ) if the corresponding Lagrangian plane is given by where the vector fields D i and ∂ pij are tacitly assumed to be evaluated at m 1 .
Remark 1. We always use the symbol D i for the truncated total derivative with respect to x i , i = 1, 2, the order of truncation depending on the context. For instance, the order of truncation is 0 in (15) and it is 1 in (18) and (19). It is convenient to set (20) ξ where the total derivatives appearing in (20) are truncated to the (k − 1) st order. Indeed, both (17) and (19) simplify as (21) L m k = ξ 1 , ξ 2 , k = 1, 2, and their dual as L * m k = dx 1 , dx 2 , respectively. A generic point of M (resp., M (1) , M (2) ) is denoted by m (resp., m 1 , m 2 ). As a rule, when both m 1 and m 2 (resp. m and m 1 ) appear in the same context, the former is always the π 2,1 -image (resp., the π-image) of the latter. In compliance with the geometric understanding of PDEs in the context of jet spaces (see (22) below), by a k th order PDE we always mean a sub-bundle E ⊆ M (k−1) of codimension one whose fiber E m k−2 at m k−2 is henceforth assumed, without loss of generality, to be connected and with compact closure in M (k−1) . We retain the same symbol L for the tautological bundle L| E −→ E restricted to E.
We conclude this section by observing that, locally, M (k) is the k + 1 st jet-extension of the trivial bundle R 2 × R → R 2 , so that the reader more at ease with jet formalism may perform the substitution  (1) , m k ∈ M (k) . The prolongation to M (k) of a local contactomorphism ψ of M is denoted by ψ (k) . 2.2. The meta-symplectic structure on C 1 . We point out that the canonical bundle epimorphism θ glob : T M −→ T M C can be regarded as a "global" analog of the local form θ defining C, in the sense that C = ker θ glob everywhere. However, even though T M C is a rank-one bundle, it is, in general, nontrivial, so that, strictly speaking, θ glob cannot be considered as a 1-form. Similarly, the (local) conformal symplectic structure ω = dθ| C admits a "global" analog, namely (23) ω glob : C ∧ C −→ C C .
Indeed, using the local coordinates (15), one immediately sees that the derived distribution C is spanned by C and ∂ u , so that the quotient [C,C] C = ∂ u is again rank-one, and ω glob identifies with ω.
One of the main gadgets of our analysis is the co-restriction to (C 1 ) of the Levi form of C 1 , firstly investigated by V. Lychagin [18] as a "twisted" analog of the symplectic form (23) for the prolonged contact distribution C 1 and called, for this reason, meta-symplectic: (24) Ω : Meta-symplectic form on C 1 .
Notice that, unlike (23), the form (24) takes its values into a rank-two bundle so that, even locally, it cannot be regarded as a 2-form in the standard sense. Nevertheless, it can be used in place of dθ (1) for defining Lagrangian subspaces: indeed, a 2D π 2,1 -horizontal subspace of C 1 m 1 is Lagrangian if it is Ω-isotropic. Using the local coordinates (18), it is easy to realize that (C 1 ) is spanned by C 1 and ∂ p1 , ∂ p2 , so that the quotient (C 1 ) identifies with the latter, and (24) reads (25) Ω = dp ij ∧ dx i ⊗ ∂ pj .
As a vector-valued differential 2-form, Ω can be identified with the pair Ω ≡ (−dθ 1 , −dθ 2 ) where θ i = dp i − p ij dx j are the local 1-forms defining C 1 , i.e., the generators of θ (1) ⊂ Λ 1 M (1) . An intuitive definition of Ω in the context of k th order jet spaces can be found in Section 3 of [5].

Description of the main results
In the case of a 3 rd order PDE E, polynomial (7)  Let us stress that (26) locally describes the intrinsically defined hypersurface of M (2) Geometric definition of Boillat-type 3 rd order MAEs.
associated to a 2-form ω ∈ Λ 2 M (1) . The first result of this paper is concerned with the characteristic cone V E of the equations E ω . Namely, we show that V E can be used to reconstruct the equation itself, since there is an obvious way to "invert" the construction of the characteristic cone (9) out of a PDE E: take any sub-bundle V ⊆ PC 1 and associate with it the following subset Now, if V is regular enough, the corresponding E V turns out to be a genuine 3 rd order PDE, and examples of "regular enough" sub-bundles are provided by characteristic cones of 3 rd order PDEs themselves. In particular, it always holds the inclusion E V E ⊇ E and we call reconstructable (from its characteristics) an equation E such that Theorem 2. Any 3 rd order MAE E of Boillat-type is reconstructable from its characteristics, i.e., E = E V E .
The simplest examples of equations (28) are obtained when V = PD, where D ⊆ C 1 is a 3D sub-distribution. Even if Goursat himself never spoke explicitly of 3 rd order MAEs, we dub after him these equations to honor his idea to characterize multi-dimensional 2 nd order MAE through linear objects involving only 1 st order derivatives. In our case, the role of such a linear object is played by D, and (28) reads Geometric definition of Goursat-type 3 rd order MAEs.
which, locally, means that E can be brought either to a quasi-linear equation, or to the form E = {F = 0} with where A is as above and f ijk ∈ C ∞ (M (1) ). Furthermore, equations (30) form the sub-class of the equations (27) which are determined by 2-forms which are decomposable modulo the differential ideal generated by contact forms (see the beginning of Section 6.3).
The second result of this paper allows to characterizes Goursat-type 3 rd order MAEs in terms of their characteristic cone.
Theorem 3. Let E be a 3 rd order PDE. Then E is locally a 3 rd order MAE of Goursat-type (i.e., E is either quasi-linear or of the form E = {F = 0}, with F given by (31)), if and only if its characteristic cone V E contains an irriducible component V E I which is a 2D linear projective sub-bundle. Theorem 3 implies that E ⊆ M (2) is a Goursat-type 3 rd order MAE if and only if its characteristic cone decomposes as II is in its turn reducible, then its components PD 2 and PD 3 are linear as well and can be unambiguously characterized by PD 1 through the formula The third result of this paper finalizes our characterization of Goursat-type 3 rd order MAEs through their characteristics.
(2) E is fully nonlinear if and only if dim D v 1 = 1: in this case, if the projective sub-bundle V E II is not empty, then it cannot contain any linear irreducible component and , then either D 1 = D 2 or D 1 and D 2 are "orthogonal" in the sense of (33).
As a main byproduct of Theorem 4 we shall obtain a method for finding intermediate integrals, discussed in Section 7, while more consequences, mainly concerning classification issues, will be collected in the final Section 8. Even without a thorough grasp of the above results, the reader may appreciate Example 5 below, where a hands-on computation reveals the distributions behind some very simple equations, which will be recalled in the key moments of this paper.
Example 5. Let E be as in (1), with F given by p 111 − p 112 − 2p 122 (resp., p 122 and p 111 ). Then equation (7) reads Applying definition (9), one easily obtains that . Observe that, in the first case, equation (34) splits as and that the "horizontal components" h i (i.e., the first generators) of (35), (36) and (37) are precisely the three distinct roots of (39). Notice also that, in general, the distributions associated with the same quasi-linear 3 rd order MAE need not to be contactomorphic. Here it follows an easy counterexample. For instance, consider the equation p 122 = 0 above: its characteristic cone consists of the two distributions D 1 and D 2 = D 3 (see (38)), which are not contactomorphic. Indeed, the derived distribution

Vertical geometry of contact manifolds and of their prolongations
The departing point of our analysis of 3 rd order PDEs is to identify them with sub-bundles of the 1 st prolongation M (2) = (M (1) ) (1) of M (1) . Hence, a key role will be played by their vertical geometry, i.e., the 1 st order approximation of their bundle structure, and, in particular, by the so-called rank-one vectors, which are in turn linked to the notion of characteristics. In order to introduce these concept, we being with the vertical geometry of the surrounding bunlde, i.e., M (2) itself.

4.1.
Vertical geometry of M (k) and three-fold orthogonality in M (1) . For k ∈ {1, 2}, we define the vertical bundle over M (k) as follows: As regard to the contact manifold M , i.e. the case k = 0, even if it does not possess a naturally defined "vertical bundle", it is still equipped with the relative vertical bundle Note that the bundle V M is, in fact, a bundle over M (1) with 2D fibers V m 1 M .
Directly from the definition (40) of V M and the isomorphism (41) for k = 0 one obtains Example 7. Of particular importance will be the vertical vectors on M (2) which correspond to perfect cubes of covectors on L via the fundamental isomorphism (41). For instance, if (see Remark 1), then the corresponding vertical vector is Even if the sections of V M are not, strictly speaking, vector fields, and a such they lack an immediate geometric interpretation, the bundle V M itself has important relationships with the contact bundle C 1 . Namely, on one hand, it is canonically embedded into the module of 1-forms on C 1 (see (48) later on) and, on the other hand, it is identified with the quotient distribution (C 1 ) C 1 (Lemma 8 below). Lemma 8. There is a (conformal 4 ) natural isomorphism Proof. It follows from a natural isomorphsim between the left-hand sides of (42) and (45). Indeed, the map induced from π * is well-defined and linear, for all m 1 ∈ M (1) . In other words, (46) is a bundle morphism, and a direct coordinate approach shows that it is in fact an isomorphism.
We stress that there is no canonical way to project the bundle C 1 over the tautological bundle L, if both are understood as bundles over M (2) . Nevertheless, if C 1 and L are regarded as bundles over M (1) , then π * turns out to be a bundle epimorphism from C 1 to L, i.e., Dually, epimorphism (47) leads to the bundle embedding L * → C 1 * which can be combined with the identification L * ∼ = V M . The result is a (conformal) embedding of bundles over M (1) , which will be useful in the sequel. In local coordinates, (48) reads ∂ pi | m −→ d m 1 x i , i = 1, 2. Now we are in position to define the concept of orthogonality in the meta-symplectic context (that has apparently never been observed before), which generalizes the "symplectic orthogonality" within the contact distribution C of M . Indeed, an immediate consequence of Lemma 8 is that the meta-symplectic form Ω is L * -valued, i.e., In turn, thanks to the canonical projection (47) of C 1 over L, the form (49) descends to a trilinear form Consider for a moment a 4D linear symplectic space (V, ω). Two vectors v 1 , v 2 ∈ V are orthogonal if ω(v 1 , v 2 ) = 0 and the orthogonal complement L ⊥ of a given 2D subspace L ⊆ V is defined by the system of two equations ω(L, · ) = 0, L being Lagrangian if L = L ⊥ . Such notions (orthogonal vectors, orthogonal complement, Lagrangian subspaces) can be found also in a 5D meta-symplectic space, but with more ramifications. For example, there can be up to two distinct, so to speak, "orthogonal complements" to a given subspace.
Remark 10. In this perspective, condition (33) expresses precisely the fact that the pair {D 2 , D 3 } is the orthogonal complement of the distribution D 1 , in the sense that Observe that the notion of a Lagrangian sub-distribution of C 1 is richer with respect to the case of sub-distributions of C: first, the pair {D 2 , D 3 } needs not to exist, then there is the case when {D 2 , D 3 } degenerates to a singleton, and the extreme case when it is the whole triple {D 1 , D 2 , D 3 } which degenerates.
, namely those sitting in the image of the Veronese embedding PL * → PS k+1 L * , i.e., the "k th powers" of sections of L * . More geometrically, these are the tangent directions at m 2 = m 2 (0) to the curves m k (t) such that the corresponding family L m 2 (t) of Lagrangian subspaces in M (k−1) "rotates" around a common hyperplane (which, in our case, is a line).
, for some α ∈ PL * m k , in which case we call α a characteristic covector and α a characteristic (direction) in the point m k . The subspace H α := ker α ≤ L m k is called the characteristic hyperplane associated to the rank-one line . Furthermore, if ω ∈ C k * is a vertical form, a hyperplane H α is called characteristic for ω if ω| = 0.
Denomination rank-one refers to the rank of the multi-linear symmetric form on L involved in the definition (see, e.g., [10]). Observe that, in our contest, dim H α = 1, so that we shall speak of a characteristic line and call characteristic vector a generator of H α . The geometric relationship between H α and = α ⊗3 is well-known (see, e.g., [28]) and it can be rendered by Remark 12 (Lines are hyperplanes). Take the local expression (43) for α and use the basis (20) From now on, characteristic directions (which are covariant objects in M (1) ) and characteristic lines (which are contravariant objects in M (1) ) will be taken as synonyms, thus breaking the separation proposed in Table 2. It should be stressed that in the multidimensional case 5 it is no longer possible to regard the characteristic directions as lines lying in L, since the latter are not hyperplanes.

4.3.
Canonical directions associated with orthogonal distributions. It is easy to see that there are two characteristic lines H 1 and H 2 which are characteristics for a vertical form ω ∈ C 1 * which is decomposable by means of Lemma 6. The meta-symplectic form (27) links these two lines with the vertical distribution V := ker ω determined by ω.
characteristic covector Table 2: To the intuitive idea of a "characteristic line", which formalizes an infinitesimal Cauchy datum giving rise to an ill-posed Cauchy problem (see the discussion of Section 1.1), there corresponds a multitude of heterogeneous yet equivalent objects, but such a redundancy is necessary to fit the same idea into multiple contexts.
Lemma 13. It holds the following identification where " " is meant via (48). In particular, dim V Ω Hi = 1, i = 1, 2. Proof. We shall assume that the coefficient of dp 11 is not zero: other cases can be dealt with likewise. So, being ω decomposable, it can be brought, up to a scaling, to the form ω = dp 11 − (k 1 + k 2 )dp 12 + k 1 k 2 dp 22 ∈ C * . Accordingly, (41)) is chosen, then the factor of (54) with respect to H i is the line Recall now that in the case of a classical, i.e., 2 nd order MAE E D determined by D = X 1 , X 2 ⊂ C, the annihilator of D ⊥ is described by and, moreover, the 2-form ω = X 1 (θ) ∧ X 2 (θ) is such that E ω = E D (see [2]). Proposition 32 later on will generalize (56) to the case of 3 rd order MAEs, where there can be up to three mutually "orthogonal" distributions, in the sense of Definition 9. Lemma 13 is the key to achieve such a generalization since it implies that, associated with a quasi-linear MAE, whose symbol is completely decomposable, there are three canonical lines in M .
Example 14. Let E be the first equation of Example 5. The following three lines are canonically associated with the triple (D 1 , D 2 , D 3 ), i.e., with the equation E. It is worth observing that the vertical part D v 1 is an integrable vertical distribution on M (1) , so that, in this case, lines (58) and (59) are the characteristic lines of the family of equations 2p 22 − p 11 + p 12 = k(x 1 , x 2 , u). Nevertheless, integrability D v 1 is not indispensable for associating directions (58) and (59) with the distribution D 1 , since they occur as the characteristic lines of the (not necessarily closed) vertical covector 2dp 22 − dp 11 + dp 12 , which annihilates D v 1 .

Characteristics of 3 rd order PDEs
Now we turn our attention to the hypersurfaces in M (2) that play the role of 3 rd order PDEs in our analysis.
if the corresponding rank-one line is tangent to E m 1 in m 2 , in which case α is a characteristic (direction) for E in m 2 and the subspace H α is a characteristic hyperplane for E.
Formula (51) says precisely that the rank-one line which corresponds to α ⊗3 via the isomorphism (41) is precisely the (one-dimensional) tangent space to the prologation H (1) α . In this perspective, α is a characteristic for E at m 2 if and only if H (1) α is tangent to E m 1 at m 2 but, in general, H α does not need to touch E m 1 in any other point.
Example 17. Function (1) determines, for k = 2, the 3 rd order PDE E = {F = 0}. Now equation (7) can be correctly interpreted as follows: it is satisfied if and only if the vector ν = ν i ξ i given by (8) spans a characteristic line of E. The very same equation (7) tells also when the covector α = ν 2 dx 1 − ν 1 dx 2 spans a characteristic direction of E. In the last perspective, equation (7) is nothing but the right-hand side of (44) applied to F and equated to zero. 5.1. Characteristics of a 3 rd order PDE and relationship with its symbol. For any m 2 ∈ E m 1 we define the vertical tangent space to E at m 2 as the subspace called the symbol of E by many authors (herewith we prefer to use the term "symbol" only for the function determining E). Obviously, vertical tangent spaces can be naturally assembled into a linear bundle called the vertical bundle of E. Directly from (60) it follows the bundle embedding V E ⊆ V M (2) E , where fibers of the former are hyperplanes in the fibers of the latter. Thanks to Lemma 6, V m 2 E can also be regarded as a subspace of S 3 L * m 2 , and, being the identification (41) manifestly conformal, such an inclusion descends to the corresponding projective spaces, i.e., where ξ i has been defined in (20). Observe that Ann (V m 2 E) is independent of the choice of F in the ideal determined by E, so that Ann (V E) can be replaced with Smbl F .
In view of (64), a direction α ∈ PL * m 2 is a characteristic one for E at m 2 if and only if (68) where · , · is the canonical pairing on S 3 L m 2 . Needless to say, (68) is independent of the choice of α (resp., F ) representing α (resp., E). Similarly, which eventually clarifies formula (7).

5.2.
The irreducible component V E I of the characteristic cone of a 3 rd order PDE E. Regard L as a sub-bundle of C 1 bearing in mind the diagram Let E = {F = 0} and identify Ann (V E) with Smbl F as in Remark 18. As a homogeneous cubic tensor on L (see (65)), Smbl F possesses a linear factor, i.e., a section In other words, if (74) is satisfied, then α ∈ PL * m 2 is a characteristic direction (incidentally revealing that 3 rd order PDEs always possess a lot of them).
Moreover, in view of the identification of characteristic directions with characteristic lines (Remark 12), (74) shows also that the line Smbl I,m 2 F ∈ PL m 2 is always a characteristic line for E, inasmuch as there always is an α such that (74) is satisfied.
It is clear now how diagram (70) can be made use of and how to define, much as we did in Section 4.2, a subset V E I ⊆ V E fitting into the commutative diagram parallel to (79). This gives a solid background to the statement of Theorem 3, where V E I was mentioned without exhaustive explanations.
5.3. The characteristic variety as a covering of the characteristic cone. Now we can come back to the direction ν defined by (8) and notice that it lies in L m 2 , and also provide an intrinsic way to check whether ν is characteristic or not. Indeed, in view of the fundamental isomorphism (41), one can regard the cube ν ⊗3 of ν as a tangent vector to the fiber M (2) m 2 , and equation (7) tells precisely when ν ⊗3 belongs to the sub-space V m 2 E = T m 2 E m 1 (see (60)), up to a line-hyperplane duality (see also Remark 12). In other words, the set of characteristic lines is the projective sub-variety in PL * m 2 cut out by (7). Fig. 1 displays in green the characteristic lines, inscribed into the corresponding fibers of the tautological bundle.
Since we plan to carry out an analysis of certain PDEs via their characteristics, it seems natural to consider all characteristic lines at once, i.e., as a unique geometric object. Traditionally, one way to accomplish this is to take the disjoint union known as the characteristic variety of E (see, e.g., [9]). As a bundle over E, the family of the fibers of char R E coincides with the equation E itself. So, the bundle char R E, in spite of its importance for the study of a given equation E, cannot be used to define E as an object pertaining to M (1) . Still this can be arranged: it suffices "to project everything one step down", so to speak. The result is precisely the characteristic cone V E : As revealed by the choreography of Fig. 1, some characteristic lines, which are distinct entities in char R E, may collapse into V E (e.g., when the corresponding fibers of the tautological bundle have nonzero intersection): hence, V E it is not the most appropriate environment for the study of characteristics. On the other hand, V E is a bundle over M (1) , where there is no trace of the original equation E: as such, it may effectively replace the equation itself and serves as a source of its invariants. Now we can give a rigorous definition of our main tool, the characteristic cone of E. Indeed, thanks to Remark 12, the characteristic variety char R E can be regarded as a sub-bundle of PL −→ E. In turn, this makes it possible to use the commutative diagram (70) to map char R E to PC 1 and define the sub-bundle V E ⊆ PC 1 as the image of such a mapping.
By its definition, V E fits into the commutative diagram revealing that, in a sense, V E is covered by char R E (see again Fig. 1). It is worth observing that, from a local perspective, (9), (78), and Definition 22 all define the same object. For instance, by Definition 22, a line H = ν ∈ PC 1 m 1 belongs to V E m 1 if and only if there is a point m 2 ∈ E m 1 such that H is a characteristic line for E at the point m 2 . But Example 17 shows that this is the case if and only if the generator ν = (ν 1 , ν 2 ) of H satisfies equation (7).

Proof of the main results
Now we are in position to prove the main results, Theorem 2, concerned with the structure of Boillat-type 3 rd order MAEs, Theorem 3 and Theorem 4, concerned with the structure of Goursat-type 3 rd order MAEs. Since the former are special instances of the latter, we prefer to clarify their relationship beforehand. 6.1. Reconstruction of PDEs by means of their characteristics. In spite of its early introduction (9), the main object of our interest has been defined again in a less direct way, which passes through the characteristic variety (see above Definition 22). The reason behind this choice is revealed by diagram (79): the characteristic variety char R E plays the role of a "minimal covering object" for both the equation E itself and its characteristic cone V E . This situation is common to various area of modern Mathematics and goes under different names, depending on the context (Bäcklund transformation, double fibration transform, Penrose transform, etc. 6 ), and it often boils down to the introduction of a suitable incidence correspondence. In the geometric theory of PDEs it is easy to guess 7 that such a correspondence must be defined in terms of Ω-isotropic flags on M (1) , in the sense of Section 2.2, i.e. by means of the commutative diagram 6 See also [14] on this concern. 7 In spite of its plainness, there is no trace of such a structure in the classical literature. Apparently, it is the second author who first introduced it explicitly [23].
H} is the flag bundle of Ω-isotropic elements of C 1 . The "double fibration transform" associated to the diagram (80) allows to pass from a sub-bundle of M (2) (e.g., a 3 rd order PDE E) to a sub-bundle of PC 1 (e.g., its characteristic cone V E ), and vice-versa. Lemma 23 indicates how to proceed in one direction.
Lemma 23. Let E be a PDE.
(1) Take the pre-image −1 2 (E) of E and select the points of tangency with the 1 -fibers: the result is char R E.
Proof. Item 1 follows from the fact that the 1 -fibers are the prolongations H (1) of hyperplanes H ∈ PC (1) , and the tangency condition means that they are determined by characteristics (see (51)). Item 2 is just a paraphrase of Definition 22.
Coming back is easier. Indeed, given a sub-bundle V ⊆ PC 1 , the same definition (28) of E V given earlier can be recast as In spite of the name "double fibration transforms", performing (81) first, and then applying Lemma 23 to the resulting equation E V , does not return, as a rule, the original sub-bundle V. We stress that, for any odd-order PDE E, it holds the inclusion This paper begins to tackle the problem of determining those PDEs which are reconstructable from their characteristics in the sense that (29) is valid. Lemma 25 below provides a simple evidence that, as a matter of fact, not all PDEs are reconstructable.
Lemma 25. V is made of strongly characteristic lines for E V .
Proof. Let H ∈ V be a line belonging to V. Then 2 ( −1 1 (H)) is nothing but H (1) (see Definition 16) and (81) tells precisely that H (1) is entirely contained into E V . Hence, H is a strongly characteristic line for E V in any point of 2 ( −1 1 (H)).
The meaning of Theorem 3 is that 3 rd order MAEs of Goursat-type are, in a sense, those PDEs whose characteristic cone takes the simplest form, namely that of a linear projective sub-bundle, and, moreover, they are also reconstructable from it. Observe that (30) is a particular case of (81), so that Goursat-type 3 rd order MAEs constitute a remarkable example of equations determined by a projective sub-bundles of PC 1 . Then Lemma 25 reveals that all the lines lying in D are strongly characteristic lines for E D , in strict analogy with the classical case [2].
Example 26. In this example we shall use a different notation from the rest of the paper. Let L = x, y be a 2D real vector space, L * = ξ, η its dual, and ω := ξ ∧ x + η ∧ y the canonical symplectic form on V := L ⊕ L * . Regard S 2 L * as the open and dense subset of the Lagrangian Grassmannian L(V, ω) which is made of planes transversal to L * , i.e., the generic fiber of the bundle J 2 (2, 1) → J 1 (2, 1). In this perspective, a quasi-linear 2 nd order PDE in two independent variables corresponds to a hyperplane where q ∈ (S 2 L * ) * = S 2 L is a symmetric tensor on L, understood as a linear map on S 2 L * . The "linear" analog of diagram (80) reads where L 2,1 (V, ω) is the manifold of ω-isotropic flags of type (2, 1). It is possible to prove that the equation E given by (84) is the double-fibration transform of PQ ⊆ PV , where Q := {Q = 0} is a quadric hypersurface in V , canonically associated with q, i.e., where E PQ is the "linear" analog of (81). We sketch (see also [15]) the procedure to obtain Q ∈ S 2 V * just by using q and ω. First, regard q as a homomorphism L * q −→ L, and take its adjoint L q * −→ L * . Second, observe that q + q * can be seen both as an endomorphism of V and as an endomorphism of V * , so that the diagram It is easy to verify that the commutator Q := [q + q * , ω] provides us with the sought-for quadratic form on V . In particular, to realize that Q is symmetric, just write down in the above basis, and observe that for any choice of L m 2 . Take now the tangential map of (90), After some obvious identifications, (91) becomes a linear map S 3 L * m 2 −→ S 3 L m 2 , i.e., an element of the tensor square (S 3 L m 2 ) ⊗2 which can be projected, via symmetrization, over S 6 L m 2 . Hence, letting m 2 vary, we obtain a section Ann V E : M (2) −→ P(S 6 L) representing a sort of "vertical covariant derivative" of (65). By the same reasons explained in Remark 18, we can safely identify Ann V E with The section Smbl F conformally defined by (92) above can be called the (symmetric) derivative of the symbol of E, and a direct computation (see [7,8]) shows that E is locally of Boillat-type if and only if Smbl F = 0.
Fix now a point m 1 ∈ M (1) and recall (see Lemma 6) that the fiber M (2) m 1 is an affine space modeled over S 3 L * m 1 . Let ω ∈ Λ 2 C 1 * and regard the corresponding 2-form ω m 1 on C 1 m 1 as a linear map Λ 2 C 1 is the so-called hyperplane section determined by ω m 1 . Now we show that H corresponds precisely to the fiber E ω,m 1 of the Boillat-type 3 rd order MAE determined by ω, via the Plücker embedding. Indeed, S 3 L * m 1 is embedded into the space L * m 1 ⊗S 2 L * m 1 of all, i.e., not necessarily Lagrangian, 2D horizontal subspaces, via the Spencer operator. In turn, L * m 1 ⊗ S 2 L * m 1 is an affine neighborhood of L m 1 in the Grassmannian Gr (C 1 m 1 , 2), which is sent to PΛ 2 C 1 m 1 by the Plücker embedding. Hence, the subspace E ω,m 1 of M (2) m 1 can be regarded as a subspace of PΛ 2 C 1 m 1 , and (27) tells precisely that such a subspace coincides with H defined by (93).
We are now in position to generalize a result about classical MAEs ( [2], Theorem 3.7) to the context of 3 rd order MAEs.
Proposition 28. A characteristic direction for E ω is also strongly characteristic.
Proof. Let H ∈ PC 1 m1 be a characteristic direction for E ω at the point m 2 . This means that H ⊂ L m 2 and that the prolongation H (1) determined by H is tangent to E ω,m 1 at m 2 . We need to prove that the whole H (1) is contained into E ω,m 1 .
To this end, recall that H (1) ⊂ M (2) m 1 is a 1D affine subspace modeled over S 3 Ann H, passing through L m 2 (see, e.g., the proof of Theorem 1 in [5]). By the above arguments, H (1) can be embedded into PΛ 2 C 1 m 1 as well. Now we can compare H (1) and H: they are both linear, they pass through the same point L m 2 , where they are also tangent each other. Hence, H (1) ⊂ H.

Now Proposition 28 allows us to prove Theorem 2.
Proof of Theorem 2. Inclusion (82) is valid for any 3 rd order PDE. Conversely, if m 2 ∈ E V E , then there is a line H ∈ V E , such that L m 2 ⊃ H. But H is a strong characteristic line for E thanks to Proposition 28, so that all Lagrangian planes passing through H and, in particular, L m 2 ≡ m 2 itself, belong to E.
6.3. Proof of Theorem 3. As it was outlined in Section 3, Goursat-type MAEs are the Boillat-type MAEs which correspond to decomposable forms, modulo a certain ideal. Before proving Theorem 3, we make rigorous this statement. To this end, we shall need the submodule of contact 2-forms and the corresponding projection Observe that the quotient bundle Λ 2 M (1) Θ is canonically isomorphic to the rank-one bundle Λ 2 L * over M (1) . Hence, (95) can be thought of as (Λ 2 L * )-valued.
Remark 29. Two 2-forms have the same projection (95) if and only if they differ by an element of Θ.
Next Proposition 31 provides an interesting link between the (complete) decomposability of the symbol of a Boillat-type MAE E and the decomposability of the corresponding two-form. Even if its proof relies on Theorem 4, which will be proved later one, this is the appropriate moment to present it.
Proposition 31. It E = {F = 0} is a 3 rd order MAE of Boillat-type whose symbol Smbl F is completely decomposable, then E is quasi-linear of Goursat-type.
Proof. Let V be the characteristic cone of E, and V I an its irreducible component. Then, by Theorem 2, E = E V . Observe also that if m 2 ∈ E V , then L m 2 contains, in particular, a line H ∈ V I , since each irreducible component of V corresponds to one of the factors of Smbl F . It follows that E = E VI as well (this fact will be proved for all Goursat-type 3 rd order MAEs in Corollary 36).
If we prove that V contains (at least) two irreducible components V I which are linear, then, by Theorem 4, E must be of the form E D and, by the same Theorem 4, also quasi-linear. To this end, notice that, for any point m 1 , the restriction Smbl F | M (2) m 1 is polynomial in the p ijk 's of degree at most one. Hence, at least two out of it its three factors must be constant, and they correspond precisely to the sought-for linear components of V.
Observe also that equation (120) may be thought of as the condition for the function G corresponding to a local Boillat-type 3 rd order MAE (26) to give a Goursat-type MAE.
Before starting the proof of Theorem 3, we provide a meta-symplectic analog of the well-known formula Ann D ⊥ = D ω.
where H 2 and H 3 are the characteristic lines of D v 1 . Proof. Let D 1 = aD 1 + bD 2 , X 1 , X 2 , with X 1 and X 2 as in (53). Then, the quasi-linear 3 rd order MAE determined by In order to obtain the right-hand side of (99), compute first aD 1 + bD 2 Ω = (adp 11 + bdp 12 ) ⊗ ∂ p1 + (adp 12 + bdp 22 ) ⊗ ∂ p2 , and factor it by H i , Combining (100) above with (55), yields Finally, the wedge product of the two 1-forms spanning the module (101) above, is the 2-form ω = ak j dp 11 ∧ dx 1 + (b − k i a)k j dp 12 ∧ dx 1 − bk i k j dp 22 ∧ dx 1 + adp 11 ∧ dx 2 + (b − k i a)dp 12 ∧ dx 2 − k i bdp 22 ∧ dx 2 , and direct computations show that E ω = E D1 . The result follows from Proposition 30. Now we turn back to Theorem 3, and deal separately with its two implications.
6.3.1. Proof of the sufficient part of Theorem 3. The sufficient part of Theorem 3 will be proved through Lemma 33 below, which is purely local, that is, all objects involved are defined on the domain of a local chart.
Proof. Let us begin with the fully nonlinear case. In order to verify that E = E D , it suffices to put where (R, S, T ) = A is the same appearing in (31), and observe that (see also (19)) We now prove that V E = PD. First of all, solve equation (31) with respect to p 111 and take the remaining coordinates p 112 , p 122 , p 222 as local coordinates on E. Thus Smbl F (see (67)) is, up to a factor, equal to so that (see also Section 5.2) A direct computation shows that so that V E I turns out to be the 3D linear space (103). Let us now pass to the quasi-linear case. One of the coefficients of the third derivatives of (102) must be nonzero. Assume that a = 0; the remaining cases can be treated similarly. Again, we solve equation (102) with respect to p 111 , so that p 112 , p 122 , p 222 become local coordinates on it, and (108) where q(ξ 1 , ξ 2 ) is a homogeneous quadratic function in ξ 1 and ξ 2 . Thus, we have that (see again Section 5.2). (109) Let us assume d = 0. In order to verify that E = E D , it is enough to put To prove that V E I = PD, observe that the symbol (108) contains the linear factor ξ 1 , and is the projectivization of (110). Finally assume d = 0.
Observe that the lines in V E I , which, in view of (109), are generated by p 112 , p 122 , p 222 ∈ R, fill a 3D linear space since ξ1 ξ2 = − h k is a solution to (108), i.e., So, if we put (112) with h, k = 0 such that (111) is satisfied, we obtain E = E D and V E I = PD.

6.3.2.
Proof of the necessary part of Theorem 3. Being E m 1 a closed submanifold of codimension 1, in the neighborhood of any point m 2 ∈ E m 1 , we can always present E in the form E = {F = 0}, with F := p ijk − G, with G not depending on p ijk , for some (i, j, k) which we shall assume equal to (1, 1, 1), since the other cases are formally analog. Then the symbol of F at m 1 is The right-hand side of (113) is a 3 rd order homogeneous polynomial with unit leading coefficient: hence, there exist unique β, A, B such that Smbl m 1 F = (ξ 1 + βξ 2 )(ξ 2 1 + Aξ 1 ξ 2 + Bξ 2 2 ). Following the general procedure (see also formula (78) and Definition 22), to construct the characteristic cone V E , one easily sees that V E contains the following 3-parametric family of lines: Observe that, with the same notation as (8), the direction v = (v 1 , v 2 ) belongs to V E I if and only if there exists m 2 ∈ E m 1 such v 1 + β(m 2 )v 2 = 0 (see also Remark 12).
Then, in the points with ∆(p 122 , p 222 ) = 0 one can find G as a polynomial expression of p 112 , p 122 , p 222 , whose coefficients turn out to be minors of the matrix (119). In particular, p 111 − G can be singled out from the so-obtained expression, namely We need to show that (120) is satisfied if and only if there is a nowhere zero factor λ such that 6.4. Proof of Theorem 4. It will be accomplished in steps. As a preparatory result, we show that a 3D sub-distribution of C 1 with 2D vertical part determines a quasi-linear 3 rd order MAE (Lemma 34 below). The converse statement, i.e., that the characteristic cone of a generic quasi-linear 3 rd order PDE possesses a linear sheet determined by a 3D sub-distribution D ⊆ C 1 with 2D vertical part, has been proved by Lemma 33 above. Then we pass to generic 3D sub-distributions of C 1 , and derive the expression of the corresponding equation E D (Lemma 35). As a consequence of these results (Corollary 36), we prove the initial statement of Theorem 4. The next three sections 6.4.1, 6.4.2, 6.4.3 are devoted to the specific proofs of the items 1, 2, 3 of Theorem 4, respectively. Besides the proof of Theorem 4 itself, a few interesting byproducts will be pointed out. For instance, Corollary 36 means that for quasi-linear 3 rd order MAEs, all characteristic lines are strong, generalizing an analogous result for classical MAEs (see [2], Theorem 3.7). Also Corollary 39 is a nontrivial and unexpected generalization of a phenomenon firstly observed in the classical case (see [2], Theorem 1.1).
Observe that dim Gr (C 1 , 3) = 6, i.e., it takes 6 parameters to identify a 3D sub-distribution D ⊂ C 1 and, hence, the corresponding equation E D .
Lemma 35. The local form of a generic E D with D ∈ Gr (C 1 , 3), in the case dim D v = 1 and up to a change of variables, is given by (31).
Proof. We can always find a coordinate system such that the (conformally unique) generator υ of D v is given by with RT − S 2 = 0. Indeed, if RT − S 2 = 0 in some coordinates (16), then a linear change of independent variables (x 1 , x 2 ) → ( x 1 , x 2 ) = (ax 1 + bx 2 , cx 1 + dx 2 ) (which is an obvious contact transformation) is enough to obtain, in the new coordinates, R T − S 2 = 0. So, without loss of generality, let us suppose that the generator of D v is given by (130) with R, S, T ∈ C ∞ (M (1) ) such that RT − S 2 = 0. Accordingly, distribution D can be always put in the form for any α, β ∈ C ∞ (M (1) ). Moreover, α and β can be chosen in such a way that Corollary 36. Let E be an equation with V E I = PD 1 . Then E = E D1 . Proof. Observe that any L m 2 , with m 2 ∈ E, always contains a characteristic line H corresponding to a fixed linear factor of the symbol, i.e., an element H belonging to V E I (see Section 5.2). In other words, inclusion (83) can be made more precise: It remains to prove that the inverse of inclusion (133) is valid when V E I = PD 1 , and to this end it is convenient to regard (133) as an inclusion of subsets in the compactification M (2) (see Section 5.1).
Indeed, Lemma 34 and Lemma 35 together guarantee that the left-hand side of (133) is a closed submanifold of codimension one and, hence, fiber-wise compact, so being M (2) . So, the right-hand side of (133) is a closed submanifold of codimension zero, i.e., an open and closed subset of the left-hand side, which is also connected. Hence, the two of them must coincide as well.
6.4.1. Proof of the statement 1 of Theorem 4. The proof of its first claim is contained into the next corollary.
The remainder of statement 1 is concerned with the "other component" V E II of V E , i.e., the one associated with the quadratic factor of Ann V E (see Section 5.2). Recall that V E II might be empty, in which case there is nothing to prove. At the far end, there is the case when V E II is, in its turn, decomposable, i.e., it consists of two linear sheets, which is dealt with by Lemma 38 below.

If
(135) is the 3D sub-distribution of C 1 corresponding 9 to the i th linear factor of (134) and Proof. To begin with, (134) dictates some restrictions on F , which must be of the form We begin with the "homogeneous" case, i.e., we assume c = 0 since, as we shall see at the end of the proof, the general case can be easily brought back to this one. Equating (137) to zero allows to express p 111 as a linear combination of p 112 , p 122 , and p 222 , i.e., to identify E m 1 with R 3 ≡ {(p 112 , p 122 , p 222 )}. In turn, this makes it possible to parametrize the space of vertical elements of (135) by three real parameters, viz., (138) D v i = −((k1+k2+k3−ki)p112+(k1k2+k1k3+k2k3)p122+k1k2k3p222)∂ p11 + (p112+kip122)∂ p12 + (p122+kip222)∂ p22 | p 112 , p 122 , p 222 ∈ R It is worth observing that, in compliance with Lemma 33, the dimension of D v i , which equals the rank of the 3 × 3 matrix In order to find a basis for (138), regard the matrix (139) as a (rank-two) homomorphism M i : R 3 −→ V = ∂ p11 , ∂ p12 , ∂ p22 and compute its kernel: Then, independently on i (and on the value of k i as well), R 3 = (1, 0, 0), (0, 1, 0) ⊕ ker M i , and D v i = M i · (1, 0, 0), M i · (0, 1, 0) is the sought-for basis. In other words, instead of (135), we shall work with the handier Concerning (136), introduce similar descriptions of its horizontal and vertical part: This concludes the preliminary part of the proof. Now impose condition (33): Observe that (145) consists, in fact, of two requirements, which are going to be dealt with separately.
The first one corresponds to the one-dimensionality of the subspace i.e., to the equation Interestingly enough, above coefficients (148), (149) and (150) characterize the dual direction of D v , i.e., a nonzero covector ω = Adp 11 + Bdp 12 + Cdp 22 ∈ V * , defined up to a nonzero constant, such that This dual perspective on D v allows to rewrite (147) as D v = (Ak 2 i − Bk i )∂ p11 + A∂ p22 , −B∂ p11 + A∂ p12 , which, in comparison with (144), depending on 6 parameters, needs only 2 of them, or even 1, if a nonzero constant is neglected. Now, thanks to (154), it is easier to see that the space (146) is 1D. Indeed, (146) reads where the first vector equals the second multiplied by k i , so that it can be further simplified: We can pass to the other condition dictated by (145). In particular, the subspace Above equation (157), makes it evident that, for any i = 1, 2, 3, the space (155) is 1D if and only if which corresponds to Plugging (158) into (156), we find a unique vector generating Ω(h, D v ). Introducing the complement c(i, j) of {i, j} in {1, 2, 3}, (160) reads To conclude the proof, recall that, besides their one-dimensionality, (145) also requires the equality of the subspaces (154) and (155) Thanks to (162), and (163) allows to eliminate B from (154): Being a nonzero constant, 10 A can also be removed from (164), which becomes: To enlighten the conclusions, it is useful to rewrite together (159) and (165) above: there are exactly 2 distributions D which are "compatible" (in the sense of (145)) with the distribution D i given by (135). More precisely, their horizontal and vertical parts are: respectively, for the only two possible values of j ∈ {1, 2, 3} {i}. One only needs to realize that (167) is one of the D v i 's from (142). To this end, rewrite (142) replacing i with j: , and subtract from the second vector the first one multiplied by k j : show that the coefficient of ∂ p11 in the second vector of (168) reduces to − l =j k l . Hence, (168) reads Comparing now (167) with (171) it is evident that D v must equal D v j , with j = i, and the proof of the "homogeneous" case is complete.
To deal with the general case, denote by E the equation determined by F as in (137), with c = 0, and by E c a "inhomogeneous" equation, i.e., one with c = 0. Observe that there is a natural identification i c : D 1 , D 2 −→ D 1 − c∂ p11 , D 2 between horizontal planes, giving rise to an automorphism ϕ c := i c ⊕ id V M (1) of C 1 . Easy computations show that ϕ c (D i ), i = 1, 2, 3, are precisely the three distributions associated with the factors of the symbol of E c , and plainly Ω(h, D v ) = Ω(ϕ c (h), ϕ c (D v )). So, the "inhomogeneous" case reduces to the "homogeneous" one, which has been established above. (3) in the points of M (1) where dim D v = 2, the equation E D is quasi-linear and the set (175) contains three elements D i , i = 1, 2, 3, possibly repeated and comprising D itself, which are orthogonal each other (see Definition 9).
Proof of 2. and 3. Let D be an element of the set (175). Then, in particular, In view of Corollary 24, equality (176) implies In other words, condition (32) is fulfilled by both equations E D and E D , so that the statements 1 and 2 of Theorem 4 (see above Section 6.4.1 and Section 6.4.2) can be made use of. To this end, rewrite (177) and (178) as respectively. Because of statement 1 (reps., 2) of Theorem 4, from (179) it follows that E D is quasi-linear (resp., fully nonlinear) if and only if dim D v = 2 (resp., 1) and from (180) it follows that E D is quasi-linear (resp., fully nonlinear) if and only if dim D v = 2 (resp., 1). Summing up, dim D v = dim D v = 1, 2, and the two cases can be treated separately.
Directly from (179) and (180) we get Prove now 2. If dim D v = 1, then also dim D v = 1 and the statement 2 of Theorem 4 (see Section 6.4.2) guarantees that V E D II (resp., V E D II ) cannot contain PD (resp., P D). It follows from (181) and (182) that P D = PD. Finally prove 3. If dim D v = 2, then also dim D v = 2 and, in view of the statement 1 of Theorem 4 (see Section 6.4.1), V E D II (resp., V E D II ) is either empty, in which case P D = PD, or consists of two, possibly repeated, distributions "orthogonal" to D (resp., D).
The result presented in this section mirrors the analogous result for classical multi-dimensional MAEs (see [2],Theorem 1), but displays some new and unexpected features: the threefold multiplicity of the notion of orthogonality and the distinction of the cases according to the dimension of the vertical part.

Intermediate integrals of Goursat-type 3 rd order MAEs
Theorem 4 established a powerful link between 3 rd order MAEs of Goursat-type, i.e., nonlinear PDEs of order three, and 3D sub-distributions of C 1 , i.e., linear objects involving (at most) second-order partial derivatives. Besides its aesthetic value, such a perspective also allows to formulate concrete results concerning the existence of solutions, as the first integrals of D can be made use of in order to find intermediate integrals of E D , along the same lines of the classical case (see [2], Section 6.3). Proposition 43 is the main result of this last section, showing that, for a Goursat-type 3 rd order MAE E, the notions of an intermediate integral of E and of a first integral of any distribution D such that E D = E are actually the same. To facilitate its proof, we deemed it convenient to introduce equations of the form E = E V where V = PĎ, with, for the first time in this paper, dimĎ = 2. By analogy with (30), we still write EĎ instead of E PĎ , but we warn the reader that, unlike all the cases considered so far, EĎ is actually a system of two independent equations (see Lemma 41 below).
Recall that a function f ∈ C ∞ (M (1) ) determines, in the neighborhood of an its nonsingular point, the hyperplane distribution ker df on M (1) , each of whose leaves is identified by a value c ∈ R. Following the same notation as [2], we set Proof. To simplify the notations, let E := EĎ. Then, fix m 1 ∈ M (1) and observe that E m 1 = {m 2 ∈ M (2) | L m 2 ∩Ď = 0} is a 2D manifold. Indeed, up to a zero-measure subset, E m 1 is a rank-one bundle over the 1D manifold PĎ, its fibers being the prolongations (or "rays") of the lines lying inĎ (see [11], Proposition 2.5).
To prove nonlinearity of E, write downĎ in local coordinates asĎ = a i D i + b i D i + R i ∂ p11 + S i ∂ p12 + T i ∂ p22 | i = 1, 2 and observe that E is given by the vanishing of the five 4 × 4 minors of the matrix Tedious computations show that the only cases when such minors are quasi-linear, is that either whenĎ fails to be two-dimensional or whenĎ is vertical, which are forbidden by the hypotheses.
Corollary 42. LetĎ ⊂ C 1 be a 2D sub distribution contained into ker df . Then for any Proof. Towards an absurd, suppose that and set c := f (m 1 ). Then (184) means that, over the point m 1 , which is impossible, since the left-hand side of (185) is linear while its right-hand side is either empty (in the case wheň D ⊆ V M (1) ) or nonlinear, thanks to above Lemma 41.
Proposition 43. Let E be a Goursat-type 3 rd order MAE and f ∈ C ∞ (M (1) ). and condition df | L m 2 = 0 means precisely that the 2D subspace L m 2 of C 1 is also contained into ker df , i.e., L m 2 lies in the 4D subspace C 1 ∩ ker df of C 1 . Because of (186), the 3D subspace D is also contained into C 1 ∩ ker df and, as such, it cannot fail to nontrivially intersect L m 2 . This means that m 2 ∈ E D , and (183) follows from the arbitrariness of m 2 . Suppose, conversely, that f is an intermediate integral of E. In view of (187) and inclusion (183), one has that (188) L m 2 ⊆ ker df ⇒ L m 2 ∩ D = 0 for any distribution D such that E D = E. We wish to show that, there is at least one among these distributions such that f is an its first integral, i.e., that the first inclusion of (186) is satisfied. Towards a contradiction, suppose that no distribution D is contained into ker df , i.e., thatĎ f := D ∩ker df is a 2D sub-distribution of the 4D distribution C 1 ∩ker df , for all distributions D such that E D = E. Then, by Corollary 42, it is possible to find a (2D subspace) L m 2 which is contained into C 1 ∩ ker df and also trivially intersects oneĎ f , thus violating (188).
Corollary 44. If the derived flag of D never reaches T M (1) , then E D admits an intermediate integral.

Perspectives
The correspondence between 3D sub-distributions of C 1 and Goursat-type 3 rd order MAEs established in Section 6 above is a convenient departing point for a classification of such equations, which is a problem beyond the scope of this paper. We stress that in all the key moments of our reasonings we never made use of coordinates, and we relied them only for clarifying the results and laying down examples. In other words, the framework we proposed is well-defined up to contactomorphisms. In particular, for Boillat-type MAEs, this means that So, a sufficient condition for Goursat-type 3 rd order MAEs to be contactomorphic is that so are the corresponding 3D sub-distributions of C 1 , but, as showed by Example 5, the condition needs not to be necessary too. Nevertheless, formula (190) allows one to find some convenient normal forms for a class of (generally) nonlinear PDEs through the analysis of certain objects canonically associated with them, which display two evident advantages: they are linear and also involve lower-order derivatives.
In view of the three-fold orthogonality, some precautions must be taken in facing equivalence problems. Indeed, if a contact-invariant property P is possessed by a distribution D determining E = E D , then any equation E which is contact-equivalent to E must be determined by a distribution D, having the same property P.
Inequivalence of rank-one vectors. 3D sub-distributions of C 1 possess some evident invariants, like, e.g., the dimension of the vertical part and, in the case of 1D vertical parts, their rank. This reflects on the corresponding PDEs.
For instance, within the class of fully nonlinear Goursat-type 3 rd order MAEs, equations of the form E = E D , with D = H, v , are distinguished according to the rank of v, which can be 1 or 2. Furthermore, the meta-symplectic form Ω provides a tool to distinguish between two equations whose distributions have both rank-one vertical part. Namely, a rank-one vector v is not always conctactomorphic to ∂ p11 . Indeed, independently on the horizontal 2D component H of D, (191) dim Ω(H, ∂ p11 ) = 1, whereas the rank-one vector a 2 ∂ p11 + 2ab∂ p12 + b 2 ∂ p22 , with a, b = 0, is such that above dimension (191) is 2.
Multiplicity of distributions. Within the class of quasi-linear Goursat-type 3 rd order MAEs E, the fact that there exists D such that E = E D and the dimension of the space Ω(h, D v ) is one or two, with D = h, D v , is a contact invariant of the equation. More precisely, dim Ω(h, D v ) = 1 (resp., 2) ⇔ D has multiplicity ≥ 2 (resp., 1).
In order to prove (192), let h and D v be as in (124) and (125) Rank-one lines in D v . One can further distinguish quasi-linear equations according to the integrability of D v , and also by the number of rank-one lines contained into D v , which is a contact invariant. For instance, the two integrable distributions ∂ p11 , ∂ p22 , ∂ p11 , ∂ p12 cannot be contactomorphic. Indeed, a∂ p11 + b∂ p22 is a perfect square if and only if a = 0 or b = 0, whereas a∂ p11 + b∂ p12 aξ 2 1 + bξ 1 ξ 2 is a perfect square if and only if b = 0. We stress that from the non-equivalence of distributions does not follow the non-equivalence of the corresponding equations, as shown by Example 5.
Conclusions. Summing up, any PDE E appeared so far had two remarkable properties: (i) E is of 3 rd order; (ii) characteristics of E are also strongly characteristics. Above properties guaranteed, respectively, that (i) each Lagrangian plane in E possesses a characteristic line (see Corollary 36), i.e., it belongs to E V E ; (ii) the set of Lagrangian planes containing a line which belongs to V E is entirely contained into E (see Lemma 25), i.e., inclusions E ⊆ E V E and E V E ⊆ E, respectively. In other words, we dealt with equations which can be reconstructed from their characteristics (see Section 3).
Boillat-type and Goursat-type 3 rd order MAEs provided us with an interesting class of such equations, whose characteristic cone took a particularly simple form, especially in the case Goursat-type equations, where an its single irreducible component (which turned out to be linear) carried all the relevant information (see the statement 1 of Theorem 4).
Even if our analysis of Goursat-type 3 rd order MAEs was quite comprehensive, no clues have been found concerning the structure of the characteristic cone of Boillat-type 3 rd order MAEs, which is something worth trying. Indeed, the main Theorems 3 and 4 went deep into the structure of the characteristic cone of Goursat-type MAEs, whereas a similar investigation of Boillat-type MAEs was rather superficial and limited to Theorem 2, which just established that also these kinds of equations can be reconstructed from their characteristics.
An intrinsic formula analogous to that discussed in Example 26 is still missing in the meta-symplectic context, and it could help in facing this problem, even beyond the class of MAEs, which played the role of a sort of testing ground for the theory. It should be stressed that, the fact that we dealt with 3 rd order equations added a remarkable simplification, due to the presence of a real root for the symbol, and investigating, e.g., 4 th order PDEs may turn out to be much more complicated.
Our analysis of intermediate integrals (Section 7) was limited to 2 nd order intermediate integrals, and it should be extended in order to encompass 1 st order intermediate integrals as well.
We concluded this paper with some ideas and preliminary results concerning the classification of Goursat-type 3 rd order MAEs through the properties of the corresponding characteristic cone which, in this case, reduced to a 3D distribution. Finding normal forms of such distributions along the same lines as [4] will require deeper insights on the meta-symplectic structure and it will be carried out in a forthcoming publication.