On the Darboux-Halphen system: Jacobi vs Lie

Two constructions of the Darboux-Halphen system are discussed. In the Jacobi construction we start with transcendental functions which are fixed as the first integrals. In the Lie construction we use a single-valued representation of the simple Lie algebra $sl(2,\mathbb{R})$ which is non-integrable in Lie’s terminology.