Lie theorem via rank 2 distributions (integration of PDE of class \omega=1)

TL;DR

This paper explores the geometric structure of certain PDE systems with a common characteristic, providing criteria for their integration and connecting Lie's theorem to rank 2 distributions.

Contribution

It establishes a geometric criterion for integrating overdetermined PDE systems with one characteristic, linking Lie's theorem to rank 2 distributions and methods of Darboux.

Findings

Criteria for integration in quadratures and closed form

Relation of Lie's theorem to rank 2 distributions

Discussion of nonlinear Laplace transformations

Abstract

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed form, as in the method of Darboux, and discuss nonlinear Laplace transformations and symmetric PDE models.