On Fractional-Linear Integrals of Geodesics on Surfaces

In this note, we give a criterion for the existence of a fractional-linear integral for a geodesic flow on a Riemannian surface and explain that, modulo the Möbius transformations, the moduli space of such local integrals (if nonempty) is either the two-dimensional projective plane or a finite number of points. We also consider explicit examples and discuss a relation of such rational integrals to Killing vectors. DOI 10.1134/S1061920824601836