  Suppose we are given a compact Riemannian manifold (Q,g)with completely
integrable geodesic flow. Let G be a compact connected Lie group acting freely
on Q by isometries. The natural question arises: will the geodesic flow on Q/G
equipped with the submersion metric be integrable? Under one natural
assumption, we prove that the answer is affirmative. New examples of manifolds
with completely integrable geodesic flows are obtained.
