  We study complex integrable systems on quiver varieties associated with the
cyclic quiver, and prove their superintegrability by explicitly constructing
first integrals. We interpret them as rational Calogero-Moser systems endowed
with internal degrees of freedom called spins. They encompass the usual systems
in type $A_{n-1}$ and $B_n$, as well as generalisations introduced by Chalykh
and Silantyev in connection with the multicomponent KP hierarchy. We also prove
that superintegrability is preserved when a harmonic oscillator potential is
added.
