Liouville classification of integrable geodesic flows in a potential field on two-dimensional manifolds of revolution: the torus and the Klein bottle

We study integrable geodesic flows on surfaces of revolution (the torus and the Klein bottle). We obtain a Liouville classification of integrable geodesic flows on the surfaces under consideration with potential in the case of a linear integral. Here, the potential is invariant under an isometric action of the circle on the manifold of revolution. This classification is obtained on the basis of calculating the Fomenko-Zieschang invariants (marked molecules) of the systems.