Geometry of Lie integrability by quadratures

TL;DR

This paper extends Lie theory of integration by quadratures, providing algebraic conditions for integrability of vector fields and generalizing to modules, thus broadening the class of solvable systems.

Contribution

It introduces algebraic criteria for quadrature-based integration of vector fields and generalizes the framework to modules, expanding the scope of integrable systems.

Findings

Algebraic conditions for quadrature integration of vector fields.

Generalization to modules and distributional solvability.

Broader class of integrable systems identified.

Abstract

In this paper we extend the Lie theory of integration in two different ways. First we consider a finite dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way. It turns out that the conditions can be expressed in a purely algebraic way. In a second step we generalize the construction to the case in which we substitute the Lie algebra of vector fields by a module (generalized distribution). We obtain much larger class of integrable systems replacing standard concepts of solvable (or nilpotent) Lie algebra with distributional solvability (nilpotency).