We study families of spherical metrics on the flat torus E_τ = C/Λ_τ with blow-up behavior at prescribed conical singularities at 0 and p, where the cone angle at 0 is 6π, and at p is 4π. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if the Green function G (z;τ) 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+sτ. An explicit transformation relating this family to the one with a single conical singularity of angle 6π at the origin is established in Theorem 1. 4. A rigidity result for rhombic tori is proved in Theorem 1. 5.
Kuo et al. (Fri,) studied this question.