Singular Manakov Flows and Geodesic Flows on Homogeneous Spaces

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces $SO(n)\mathrm{/}SO(k_1)\times \mathrm{.}\mathrm{.}\mathrm{.}\times SO(k_r)$, for any choice of $k_1,\mathrm{.}\mathrm{.}\mathrm{.},k_r$, $k_1+\mathrm{.}\mathrm{.}\mathrm{.}+k_r\le n$. 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 $SO(k_1+k_2+k_3)\mathrm{/}SO(k_1)\times SO(k_2)\times SO(k_3)$ and on the Stiefel manifolds $V(n,k)=SO(n)\mathrm{/}SO(k)$ is given.