  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.
