INTEGRABLE REDUCTIONS OF THE DRESSING CHAIN

Abstract

In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each k,n is an element of N with n >= 2k +1 we obtain a Lotka-Volterra system LVb(n, k) on R-n which is a deformation of the Lotka-Volterra system LV(n, k), which is itself an integrable reduction of the 2m+1-dimensional Bogoyavlenskij-Itoh system LV(2m+1, m), where m = n - k - 1. We prove that LVb(n, k) is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational first integrals of LV(n, k). We also construct a family of discretizations of LVb(n, 0), including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.