On integrability of the Kontsevich non-abelian ODE system

Details On integrability of the Kontsevich non-abelian ODE system On integrability of the Kontsevich non-abelian ODE system

2011-08-21

arxiv.org/abs/1108.4208v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

9 pages, accepted for publication in Letters in Mathematical Physics

Scientific paper

10.1007/s11005-011-0527-4

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.

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