  We consider integrable generalizations of the spherical pendulum system to
the Stiefel variety $V(n,r)=SO(n)/SO(n-r)$ for a certain metric. For the case
of V(n,2) an alternative integrable model of the pendulum is presented.
  We also describe a system on the Stiefel variety with a four-degree
potential. The latter has invariant relations on $T^*V(n,r)$ which provide the
complete integrability of the flow reduced on the oriented Grassmannian variety
$G^+(n,r)=SO(n)/SO(r)\times SO(n-r)$.
