Motivated by optimal control of affine systems stemming from mechanics, metrics on the two-sphere of revolution are considered; these metrics are Riemannian on each open hemisphere whereas one term of the corresponding tensor becomes infinite on the equator. Length minimizing curves are computed and structure results on the cut and conjugate loci are given, extending those in \cite{anihp-2008a}. These results rely on monotonicity and convexity properties of the quasi-period of the geodesics; such properties are studied on an example with elliptic transcendency. A suitable deformation of the round sphere allows to reinterpretate the equatorial singularity in terms of concentration of curvature and collapsing of the sphere.