Non-integrability of geodesic flow on certain algebraic surfaces

► The behaviour of geodesics on surfaces defined by algebraic expressions is studied. ► The non-integrability of the geodesic equations is rigorously proved using differential Galois theory. ► Morales-Ramis theory and Kovacicʼs algorithm is used and the normal variational equation is of Fuchsian type. ► Some extensions and limitations are discussed.