Fractional-linear integrals of geodesic flows on surfaces and Nakai's geodesic 4-webs

We prove that if the geodesic flow on a surface has an integral, fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for existence of fractional-linear integrals: such integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves.