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

In this paper, we investigate the following curvature equation: \begin{equation} \Delta u+e^{u}=8\pi (\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau }\text{, }\tau \in \mathbb{H} (0.1) \label{a} \end{equation} Here $E_{\tau }$ represents a flat torus and $\frac{\omega _{k}}{2}$ is one of the half periods of $E_{\tau }$. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: \begin{equation*} \Delta u+e^{u}=8\pi \delta _{0}\text{ in }E_{\tau }\text{, }\tau \in \mathbb{ H}\text{.} \end{equation*} This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all $\tau$. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: \begin{equation*} \Delta u+e^{u}=8\pi n(\delta _{0}+\delta _{\frac{\omega _{k}}{2}})\text{ in } E_{\tau },\text{ }n\in \mathbb{N}\text{.} \end{equation*} As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5.