Let $(M,[\theta])$ be a contact manifold, $\dim M=2n+1$, and denote by $\mathcal{I}^\theta$ the differential ideal generated by the contact form $\theta$.

An exterior differential system on $M$ of the form
$$
(M,\mathcal{I}^\theta+\mathcal{I})
$$
can be called a **contact EDS on $M$**.

A contact EDS on $M$ is naturally interpreted as a PDE imposed on Legendrian submanifolds of $M$: in local contact coordinates, a Legendrian (i.e., $\mathcal{I}^\theta$-integral and $n$-dimensional) submanifold is the graph of the first jet of a function $u$ in $n$ independent variables, and the condition of being also $\mathcal{I}$-integral reads as a second-order PDE on $u$. For instance, if $\mathcal{I}$ is generated by an $n$-form, then one gets a Monge-Ampere equation. (If $\mathcal{I}=0$ one gets the trivial PDE $0=0$, whose solutions are *all* the Legrendrian submanifolds.)

QUESTION: what is the class of $n$-dimensional scalar second-order PDEs which corresponds to contact EDS on $(2n+1)$-dimensional contact manifolds? Is it larger than the class of Monge-Ampere equations, or not?

I'm really curious, provided that "my" definition of contact EDS above is correct, where it can be found in the literature. (The sum $\mathcal{I}^\theta+\mathcal{I}$ is just a fancy way to say that I'm taking an ideal containing the contact ideal.)