On the Darboux-Halphen system: Jacobi vs Lie

TL;DR

This paper compares two different constructions of the Darboux-Halphen system, highlighting their mathematical differences and implications.

Contribution

It analyzes and contrasts the Jacobi and Lie approaches to constructing the Darboux-Halphen system, emphasizing their distinct features.

Findings

Jacobi construction uses multi-valued functions as first integrals.

Lie construction employs single-valued algebra representations.

The two methods differ in integrability and algebraic structure.

Abstract

Two constructions of the Darboux-Halphen system are discussed. In the Jacobi construction we start with multi-valued functions which are fixed as the first integrals. In the Lie construction we use single-valued representation of the simple algebra which is non-integrable in Lie's terminology.