  We prove complete integrability of the Manakov-type SO(n)-invariant geodesic
flows on homogeneous spaces $SO(n)/SO(k_1)\times...\times SO(k_r)$, for any
choice of $k_1,...,k_r$, $k_1+...+k_r\le n$. In particular, a new proof of the
integrability of a Manakov symmetric rigid body motion around a fixed point is
presented. Also, the proof of integrability of the SO(n)-invariant Einstein
metrics on $SO(k_1+k_2+k_3)/SO(k_1)\times SO(k_2)\times SO(k_3)$ and on the
Stiefel manifolds $V(n,k)=SO(n)/SO(k)$ is given.
