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} Δu+e^{u}=8π(δ_{0}+δ_{\frac{ω_{k}}{2}})$\text{ in }$ E_{τ}$\text{, }$τ\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*} Δu+e^{u}=8πδ_{0}$\text{ in }$E_{τ}$\text{, }$τ\in $\mathbb{H}$$\text{.}$ \end{equation*} This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all $\tau$. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: \begin{equation*} Δu+e^{u}=8πn(δ_{0}+δ_{\frac{ω_{k}}{2}})$\text{ in }$ E_{τ},$$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.