Boris Kruglikov: Cartan's 5 variables centennial: Monge equations and 2-distributions

In his celebrated paper of 1910 Elie Cartan classified Monge
equations of bi-order (1,2) with large symmetry groups and obtained the maximal symmetric model (Cartan-Hilbert equation), corresponding to the group G_2. In a joint paper with Ian Anderson we extended his classification of maximally symmetric models to Monge equations of arbitrary bi-orders. The geometry behind this is the Tanaka theory for rank 2 distributions and the algebra corresponds to interplay between central extensions of graded nilpotent Lie algebras and integrable extensions of differential systems.