The moment map on the space of symplectic 3d Monge-Ampère equations

For any 2nd order scalar PDE $\mathcal{E}$ in one unknown function, we construct, by means of the characteristics of $\mathcal{E}$, a contact sub-bundle of the underlying contact manifold J1, consisting of conic varieties, called the contact cone structure associated with $\mathcal{E}$. We then focus on symplectic Monge-Ampère equations in 3 independent variables, that are naturally parametrized, over C, by the projectivization of the 14-dimensional irreducible representation of the simple Lie group Sp(6, C). The associated moment map allows to define a rational map $\varpi$ from the space of symplectic 3D Monge-Ampère equations to the projectivization of the space of quadratic forms on a 6-dimensional symplectic vector space. We study the relationship between the variety $\varpi(\mathcal{E}) = 0$, herewith called the co-characteristic variety of $\mathcal{E}$, and the contact cone structure of a 3D Monge -Ampère equation $\mathcal{E}$, by obtaining a complete list of mutually non-equivalent quadratic forms on a 6-dimensional symplectic space.