Painlev\'e type reductions for the non-Abelian Volterra lattices

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 Painlev\'e-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 Painlev\'e equations dP$_1$ and dP$_{34}$ and for the continuous Painlev\'e equations P$_3$, P$_4$ and P$_5$.