  This is a survey on finite-dimensional integrable dynamical systems related
to Hamiltonian $G$-actions. Within a framework of noncommutative integrability
we study integrability of $G$-invariant systems, collective motions and reduced
integrability. We also consider reductions of the Hamiltonian flows restricted
to their invariant submanifolds generalizing classical Hess--Appel'rot case of
a heavy rigid body motion.
