On integrability of the Kontsevich non-abelian ODE system

TL;DR

This paper proves the integrability of Kontsevich's non-abelian ODE system by constructing a Lax pair, first integrals, and commuting flows, extending understanding of non-commutative integrable systems.

Contribution

It provides the first proof of integrability for the Kontsevich system, including explicit Lax pair, integrals, and symmetries, advancing non-commutative integrable systems theory.

Findings

Established the integrability of the Kontsevich system

Constructed a Lax pair and identified first integrals

Developed a pre-Hamiltonian operator for symmetries

Abstract

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.