Exceptionally simple PDE

We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except $\mathfrak{sl}_2$. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the $G_2$ case. We give a formula for the harmonic curvature of a $G_2$-contact structure and describe submaximally symmetric models for general $G$-contact structures.