Even cone spherical metrics: blow-up at two prescribed cone singularities

We study families of spherical metrics on the flat torus Eτ=C/Λτ with 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 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;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 1.3. A rigidity result for rhombic tori is proved in Theorem 1.4.