# Integrable reductions of the dressing chain

## TL;DR

This paper develops integrable reductions of the dressing chain in Lotka-Volterra form, introducing deformations that preserve integrability and constructing discretizations that remain superintegrable, expanding understanding of these systems.

## Contribution

It introduces a family of deformed integrable Lotka-Volterra systems and their discretizations, extending known integrable reductions of the dressing chain.

## Key findings

- The systems are Liouville and non-commutative integrable.
- Discretizations, including Kahan, are Liouville and superintegrable.
- Deformations preserve rational first integrals.

## Abstract

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