  We consider systems of ODEs with the right hand side being Laurent
polynomials in several non-commutative unknowns. In particular, these unknowns
could be matrices of arbitrary size. An important example of such a system was
proposed by M. Kontsevich. We prove the integrability of the Kontsevich system
by finding a Lax pair, corresponding first integrals and commuting flows. We
also provide a pre-Hamiltonian operator which maps gradients of integrals for
the Kontsevich system to symmetries.
