We study families of spherical metrics on the flat torus E = C/ with conical singularities at 0 and p, where the cone angle at 0 is 6, and at p is 4. We prove that the existence of a necessarily unique, even family of spherical metrics that blows up at a cone point p, is completely determined by the geometry of the torus: such a family exists if and only if the Green function G (z;) admits a pair of nontrivial critical points a. In this case, the cone point p must equal a, and the corresponding monodromy data is (2r, 2s), where a=r+s. An explicit transformation relating this family to the one with a single conical singularity of angle 6 at the origin is established in Theorem 1. 3. A rigidity result for rhombic tori is proved in Theorem 1. 4.
Kuo et al. (Wed,) studied this question.