  Let $G/K$ be an orbit of the adjoint representation of a compact connected
Lie group $G$, $\sigma$ be an involutive automorphism of $G$ and $\tilde G$ be
the Lie group of fixed points of $\sigma$. We find a sufficient condition for
the complete integrability of the geodesic flow of the Riemannian metric on
$\tilde G/(\tilde G\cap K)$, which is induced by the bi-invariant Riemannian
metric on $\tilde G$. The integrals constructed here are real analytic
functions, polynomial in momenta. It is checked that this sufficient condition
holds when $G$ is the unitary group $U(n)$ and $\sigma$ is its automorphism
defined by the complex conjugation.
