Riemannian metrics on the sphere of revolution with an equatorial singularity are introduced so as to study dynamics arising in space mechanics and two-body control. A homotopy from the round metric on the sphere is defined which gives evidence of classification by the order of singularity for such metrics. Symmetry, integrability and quasi-homogeneity properties near the singularity unveil the main features of geodesics and allow to compute asymptotics, in connection with optimality issues. This question is fully addressed in the end of the paper where cut and conjugate loci are analyzed, providing examples of bifurcation and collapsing.