The Lancet

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.