2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56873The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the bi-invariant metric on any bi-quotient of a compact Lie group is integrable in non-commutative sense by means of polynomial integrals, and therefore, in classical commutative sense by means of $C^\infty$--smooth integrals.19 pages, minor changes, to appear in Annals of Global Analysis and GeometryMathematical Physics37J35, 37J15, 70H06, 70H33, 53D20, 53D25text