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

We study families of spherical metrics on the flat torus Eτ $$=$$ C/Λ τ 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.