If one wants the ideal to represent a single second-order PDE on a single unknown function of $n$ variables well-defined up to contact transformations, then the ideal should just be $\mathcal{I}^\theta$ with a single $\mathcal{I}^\theta$-primitive $n$-form $\Upsilon$ adjoined (i.e., a Monge-Ampère system). However, one can have larger ideals than this that represent systems of second-order PDE on a single unknown function of $n$ variables. (These can be either involutive or not.)

One way to see this is to consider any hypersurface $\Sigma$ in the space $\mathcal{V}_n(\mathcal{I}^\theta)$ (which is a smooth manifold of dimension $\tfrac12(n^2{+}5n{+}2)$). You can ask when $\Sigma$ is the space of integral elements of some ideal $\mathcal{J}$ on $M^{2n+1}$, and one finds that this happens if and only if $J$ is a Monge-Ampère system.

As for involutive second-order systems that contain the contact ideal, there are many (too many to classify).

If one wants the ideal to represent a single second-order PDE on a single unknown function of $n$ variables well-defined up to contact transformations, then the ideal should just be $\mathcal{I}^\theta$ with a single $\mathcal{I}^\theta$-primitive $n$-form $\Upsilon$ adjoined (i.e., a Monge-Ampère system). However, one can have larger ideals than this that represent systems of second-order PDE on a single unknown function of $n$ variables. (These can be either involutive or not.)

One way to see this is to consider any hypersurface $\Sigma$ in the space $\mathcal{V}_n(\mathcal{I}^\theta)$ (which is a smooth manifold of dimension $\tfrac12(n^2{+}5n{+}2)$). You can ask when $\Sigma$ is the space of integral elements of some ideal $\mathcal{J}$ on $M^{2n+1}$, and one finds that this happens if and only if $\mathcal{J}$ is a Monge-Ampère system.

As for involutive second-order systems that contain the contact ideal, there are many (too many to classify).