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

For any 2ⁿd order scalar PDE E in one unknown function, we construct, by means of the characteristics of E, a contact sub--bundle of the underlying contact manifold J¹, consisting of conic varieties, called the {contact cone structure} associated with 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 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 (E)=0, herewith called the {co-characteristic variety} of E, and the contact cone structure of a 3D Monge-Ampère equation E, by obtaining a complete list of mutually non--equivalent quadratic forms on a $6$--dimensional symplectic space.