Regular and irregular geodesics on spherical harmonic surfaces

► The behavior of geodesics on surfaces defined by spherical harmonics is studied. ► The non-integrability of the geodesic equations is rigorously proved using differential Galois theory. ► Morales-Ramis theory and Kovacic's algorithm is used and the normal variational equation is of Fuchsian type. ► Poincaré sections are used to display the breakdown in integrability.