New examples of manifolds with closed geodesics (Q2266460)

In this note the author studies a family of Riemannian metrics on \(S^ 2\) such that their geodesics are all closed and the geodesic flow has an additional integral, quadratic with respect to the velocity. Let f, h be two periodic, nonnegative functions satisfying some appropriate conditions. The author considers the Riemannian metric \(ds^ 2=(f(u)+g(v))(du^ 2+dv^ 2)\) on \(S^ 2\) and obtains a system of equations satisfied by f, h under the assumption that the geodesics of the considered metric are closed.