On the completeness of the Manakov integrals

The aim of this note is to present simple proofs of the completeness of Manakov’s integrals for a motion of a rigid body fixed at a point in $\mathbb R^n$,aswellasforgeodesicflowsonaclassofhomogeneousspaces$SO(n)/SO(n_1)\times\dots\times SO(n_r)$.