Non-commutative generalization of integrable quadratic ODE systems

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.