Partial Reductions of Hamiltonian Flows and Hess-Appel'rot Systems on SO(n)

We study reductions of the Hamiltonian flows restricted to their invariant submanifolds. As examples, we consider partial Lagrange-Routh reductions of the natural mechanical systems such as geodesic flows on compact Lie groups and $n$-dimensional variants of the classical Hess-Appel'rot case of a heavy rigid body motion about a fixed point.