Solvability of a Lie algebra of vector fields implies their integrability by quadratures

TL;DR

This paper generalizes Lie's classical result by proving that all vector fields within a finite-dimensional, transitive, and solvable Lie algebra on a manifold can be integrated explicitly using quadratures, extending the understanding of integrability.

Contribution

It establishes that any finite-dimensional, transitive, and solvable Lie algebra of vector fields on a manifold is integrable by quadratures, broadening the scope of classical integrability results.

Findings

All vector fields in the specified Lie algebra are integrable by quadratures.

The result applies to finite-dimensional, transitive, and solvable Lie algebras.

Extends classical Lie integrability results to a broader class of vector fields.

Abstract

We present a substantial generalisation of a classical result by Lie on integrability by quadratures. Namely, we prove that all vector fields in a finite-dimensional transitive and solvable Lie algebra of vector fields on a manifold can be integrated by quadratures.