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

Integrable systems on multiplicative quiver varieties from cyclic quivers