  We study integrable geodesic flows on Stiefel varieties
$V_{n,r}=SO(n)/SO(n-r)$ given by the Euclidean, normal (standard),
Manakov-type, and Einstein metrics. We also consider natural generalizations of
the Neumann systems on $V_{n,r}$ with the above metrics and proves their
integrability in the non-commutative sense by presenting compatible Poisson
brackets on $(T^*V_{n,r})/SO(r)$. Various reductions of the latter systems are
described, in particular, the generalized Neumann system on an oriented
Grassmannian $G_{n,r}$ and on a sphere $S^{n-1}$ in presence of Yang-Mills
fields or a magnetic monopole field.
  Apart from the known Lax pair for generalized Neumann systems, an alternative
(dual) Lax pair is presented, which enables one to formulate a generalization
of the Chasles theorem relating the trajectories of the systems and common
linear spaces tangent to confocal quadrics. Additionally, several extensions
are considered: the generalized Neumann system on the complex Stiefel variety
$W_{n,r}=U(n)/U(n-r)$, the matrix analogs of the double and coupled Neumann
systems.
