Painlevé type reductions for the non-Abelian Volterra lattices * *To the memory of A B Shabat and R I Yamilov.

Abstract

The Volterra lattice admits two non-Abelian analogs that preserve the integrability property. For each of them, the stationary equation for non-autonomous symmetries defines a constraint that is consistent with the lattice and leads to Painlevé-type equations. In the case of symmetries of low order, including the scaling and master-symmetry, this constraint can be reduced to second order equations. This gives rise to two non-Abelian generalizations for the discrete Painlevé equations and and for the continuous Painlevé equations P3, and P5.