Integrable reductions of the dressing chain

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