  We study some classical integrable systems naturally associated with
multiplicative quiver varieties for the (extended) cyclic quiver with $m$
vertices. The phase space of our integrable systems is obtained by
quasi-Hamiltonian reduction from the space of representations of the quiver.
Three families of Poisson-commuting functions are constructed and written
explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the
tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for
$m>1$ we obtain new integrable systems that generalise the
Ruijsenaars-Schneider system. These systems and their quantum versions also
appeared recently in the context of supersymmetric gauge theory and cyclotomic
DAHAs, as well as in the context of the Macdonald theory.
