# Revisiting Lie integrability by quadratures from a geometric perspective

## TL;DR

This paper revisits Lie integrability by quadratures, analyzing conditions for integrating vector fields through quadratures, extending classical theorems with geometric insights and examples.

## Contribution

It provides a generalized geometric framework for Lie integrability, including conditions and extensions beyond classical Lie theorems.

## Key findings

- Identifies conditions for integrating vector fields via quadratures.
- Extends Lie integrability theory to distributions.
- Provides illustrative examples of the geometric approach.

## Abstract

After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way will be discussed, determining also the number of quadratures needed to integrate the system. The theory will be illustrated with examples andbn an extension of the theorem where the Lie algebras are replaced by some distributions will also be presented.