Meta-Symplectic Geometry of $3^{\mathrm{rd}}$ Order Monge–Ampère 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–Ampère 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–Ampère 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–Ampère equations, herewith called of Goursat type.