We find all homogeneous quadratic systems of ODEs with two dependent variables that have polynomial first integrals and satisfy the Kowalevski-Lyapunov test. Such systems have infinitely many polynomial infinitesimal symmetries. We describe all possible non-commutative generalizations of these systems and their symmetries. As a result, new integrable quadratic homogeneous systems of ODEs with two non-commutative variables are constructed. Their integrable non-commutative inhomogeneous generalizations are found. In particular, a non-commutative generalization of a Hamiltonian flow on the elliptic curve is presented.
Letters in Mathematical Physics
- Pub Date:
- October 2019
- Non-commutative ODEs;
- Painlevé test;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- 19 pages