Integrability of geodesic flows for metrics on suborbits of the adjoint orbits of compact groups

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.