Integrable geodesic flows and metrisable second-order ordinary differential equations

Abstract

It is well known that the system of ordinary differential equations (ODEs) describing geodesic flows of some Riemannian metrics on 2-surfaces admits a projection on a special class of second-order ODEs. In this paper we study in detail this special class of ODEs. We classify all such autonomous ODEs possessing autonomous first integrals that are fractional-quadratic in the first derivative. We construct all the families of integrable geodesic flows related to the ODEs under study. These families are parameterized by two arbitrary functions and contain metrics with superintegrable geodesic flows. We also study the inverse problem and find novel families of second-order ODEs that are related to integrable geodesic flows. We explicitly present first integrals of such ODEs. We find the general structure of metrisable second-order ODEs related to conformal metrics.