  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.
