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 Painlevé type generalizing the dP$_1$ and dP$_{34}$ equations. For these reductions, we present the isomonodromic Lax pairs and derive the Bäcklund transformations which form the $\mathbb{Z}^m$ lattice.

Comment: 17 pages. This article is part of an OCNMP Special Issue in Memory of Professor Decio Levi

• 37-XX - Dynamical systems and ergodic theory
• 37K35 - Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
• 37K60 - Lattice dynamics; integrable lattice equations
• 39A36 - Integrable difference and lattice equations; integrability tests
• 81-XX - Quantum theory