  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.
