  We consider a class of complex manifolds constructed as multiplicative quiver
varieties associated with a cyclic quiver extended by an arbitrary number of
arrows starting at a new vertex. Such varieties admit a Poisson structure,
which is obtained by quasi-Hamiltonian reduction. We construct several families
of Poisson subalgebras inside the coordinate ring of these spaces, which we use
to obtain degenerately integrable systems. We also extend the Poisson centre of
these algebras to maximal abelian Poisson algebras, hence defining Liouville
integrable systems. By considering a suitable set of local coordinates on the
multiplicative quiver varieties, we can derive the local Poisson structure
explicitly. This allows us to interpret the integrable systems that we have
constructed as new generalisations of the spin Ruijsenaars-Schneider system
with several types of spin variables.
