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 (private communication). 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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Wolf, T., Efimovskaya, O. On Integrability of the Kontsevich Non-Abelian ODE System. Lett Math Phys 100, 161–170 (2012). https://doi.org/10.1007/s11005-011-0527-4
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DOI: https://doi.org/10.1007/s11005-011-0527-4