Negative flows and non-autonomous reductions of the Volterra lattice

Abstract: We study reductions of the Volterra lattice corresponding to stationary equations for the additional, noncommutative subalgebra of symmetries. It is shown that, in the case of general position, such a reduction is equivalent to the stationary equation for a sum of the scaling symmetry and the negative flows, and is written as $(m+1)$-component difference equations of the Painlev\'e type generalizing the dP$1$ and dP${34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the B\"acklund transformations which form the $\mathbb{Z}^m$ lattice.