Sharp results for spherical metric on flat tori with conical angle 6π\pi at two symmetric points

A conjecture about the existence or nonexistence of solutions to the curvature equation (1.1) defined on a rectangle torus $E_{\tau},$ $\tau\in i\mathbb{R}_{>0}$ with four conical singularties at its symmetric points is proposed in [3]. See Conjecture 1. For the purposes to understand this problem, in this paper, we study the following equation: \Delta u+e^{u}=8\pi(\delta_{0}+\delta_{\frac{\omega_{k}}{2}})\text{in}E_{\tau}\,\tau\in\mathbb{H}\, \label{a} where $\frac{\omega_{k}}{2}$ is one of the half periods of $E_{\tau}$, i.e., the case $(m_{0},m_{1},$ $m_{2},m_{3})$ $=(1,1,0,0)$, $(1,0,1,0)$, $(1,0,0,1)$ for $k=1,2,3,$ respectively. Among others, we prove that the existence of \textit{non-even family of solutions} (see the definition in Section 1 ) is related to the existence of solutions for the equation with single conical singularity: $$\Delta u+e^{u}=8\pi\delta_{0}\text{ in }E_{\tau}\text{, }\tau\in \mathbb{H}\text{.}$$ Consequently, equation (0.1) does not have any non-even family of solutions for all $k=1,2,3$. As an application, we completely understand the solution structure of the equation (0.1) for rectangle torus and give a confirmative answer for this conjecture in this three cases. See Theorem 1.3