We study families of spherical metrics on the flat torus \(E_{\tau }\) \(=\) \(\mathbb {C}/\Lambda _{\tau }\) with conical singularities at 0 and \(\pm p\) , where the cone angle at 0 is \(6\pi \) , and at \(\pm p\) is \(4\pi \) . 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 \(G(z;\tau )\) admits a pair of nontrivial critical points \(\pm a\) . In this case, the cone point p must equal a, and the corresponding monodromy data is \(\left( 2r,2s\right) \) , where \(a=r+s\tau .\) An explicit transformation relating this family to the one with a single conical singularity of angle \(6\pi \) at the origin is established in Theorem 1.3. A rigidity result for rhombic tori is proved in Theorem 1.4.
