On the Lie noncommutative integrability

TL;DR

This paper explores the application of Lie theory to non-commutative integrability, reconstructing integrable systems in three-dimensional Euclidean space, including classical and Bianchi B class examples.

Contribution

It demonstrates how Lie theory can be used to identify and reconstruct integrable systems, highlighting specific examples like the Darboux-Brioschi-Halphen system.

Findings

Reconstruction of integrable systems using Lie theory.

Identification of the Darboux-Brioschi-Halphen system as Lie integrable.

Examples related to Bianchi B class Lie algebras.

Abstract

The Lie theory of non-commutative integrability is used to reconstruct some integrable systems of ordinary differential equations in three dimensional Eucledian space. The Darboux-Brioschi-Halphen system is an example of the Lie integrable system associated with the simple Lie algebra sl(2,R). Other examples are related with solvable three dimensional real Lie algebras of Bianchi B class.