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

TL;DR

This paper classifies rank 2 distributions of Monge equations with maximal symmetry, introduces a new Lie-Bäcklund theorem, and shows all flat Monge equations are extensions of the Hilbert-Cartan equation.

Contribution

It develops the Tanaka theory for rank 2 distributions, providing a complete classification and new insights into the structure of Monge equations.

Findings

Complete classification of Monge equations with maximal symmetry.

Introduction of a new Lie-Bäcklund theorem.

All flat Monge equations are extensions of the Hilbert-Cartan equation.

Abstract

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) pair of its orders. Investigation of the corresponding Tanaka algebras leads to a new Lie-Backlund theorem. We prove that all flat Monge equations are successive integrable extensions of the Hilbert-Cartan equation. Many new examples are provided.