Recently I. Mladenov and J. Oprea have investigated a number of surfaces of revolution, and in particular, developed numerical shooting methods to investigate geodesics on those surfaces, which in turn led them to raise some questions concerning closed geodesics on those surfaces. Here we develop explicit formulae, usually in terms of elliptic integrals, that permit us to answer the questions. The computations are based of course on the classical Clairaut's formulae, and a major point is to demonstrate that explicit computations can be made. A closed geodesic on a surface of revolution oscillates $q$ times across the equator of the surface, while winding $p$ times around the axis of rotation. Call the pair $(p,q)$ the type of the geodesic. In particular, the permitted types are explicitly determined for the investigated surfaces.
James C. Alexander. "Closed Geodesics on Certain Surfaces of Revolution." J. Geom. Symmetry Phys. 8 1 - 16, 2006. https://doi.org/10.7546/jgsp-8-2006-1-16