For any $2{^\textrm{nd}}$ 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 $J^1$, 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 ${\mathbb{C}}$, bythe projectivization of the 14--dimensional irreducible representation of the simple Lie group ${\mathsf{Sp}}(6,{\mathbb{C}})$. 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.