Rank 2 distributions of Monge equations: Symmetries, equivalences, extensions

By developing the Tanaka theory for rank 2 distributions, we completely classify classical Monge equations having maximal finite-dimensional symmetry algebras with fixed (albeit arbitrary) orders. Investigation of the corresponding Tanaka algebras leads to a new Lie–Bäcklund theorem. We prove that all flat Monge equations are successive integrable extensions of the Hilbert–Cartan equation. Many new examples are provided.