On the Lie noncommutative integrability

2026/02/18 by A. V. Tsiganov · 1 voice · 1 citation
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.DS

paper · pdf

arxiv published 2026/02/18 · arxiv updated 2026/06/25

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.

Citations

• On the Darboux-Halphen system: Jacobi vs Lie
• Solvability of a Lie algebra of vector fields implies their integrability by quadratures
• Geometry of Lie integrability by quadratures
• Generalized Liouville method of integration of Hamiltonian systems
• Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes

Cited by

• On the Darboux-Halphen system: Jacobi vs Lie

Discussions

• A. V. Tsiganov: On the Lie noncommutative integrability https://arxiv.org/abs/2602.16267 https://arxiv.org/pdf/2602.16267 https://arxiv.org/html/2602.16267 [bsky, 0 points, 0 comments]

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