Non-commutative generalization of integrable quadratic ODE systems

TL;DR

This paper classifies and constructs new integrable quadratic ODE systems with non-commutative variables, extending classical integrability results to non-commutative algebraic settings.

Contribution

It provides a complete description of non-commutative generalizations of quadratic integrable systems with polynomial first integrals.

Findings

New non-commutative integrable quadratic systems constructed

Classification of systems with polynomial first integrals and symmetries

Extension of Hamiltonian flows to non-commutative variables

Abstract

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.