Abstract: Let $E$ , $\tau \in H$ be a flat torus with periods $1$ and $\tau$, and set $w_0=0$, $w_1=1$, $w_2=\tau$, $w_e =1+\tau$. In this talk, we investigate the following curvature equation: \begin{equation}\Delta u + e^u = \sum\limits_{k=0}^3 4\pi m_k \delta_{\frac{w_k}{2}}\quad\text{on}\quad E_\tau\tag{1}\end{equation} where $m_k \in \mathbb Z$ and $\sum\limits_{k=0}^3 4\pi m_k$ is called the total curvature. This equation (1) arised originally from conformal geometry and any solution to this equation produces a conformal metric $g=\dfrac 12 e^u ds^2$ on the torus $E_\tau$, where $ds^2$ denotes the flat metric on $E_\tau$. The new metric exhibits a spherical metric but has conical singularities on each $\dfrac {w_k}2$ with angle $2\pi(1+m_k)$. When $\sum\limits_{k=0}^3 m_k$ is even, from PDE point of view, the bubbling phenomenon might happen, which makes the equation (1) become extremely difficult to study. In this talk, we will explain how to study the case $\sum\limits_{k=0}^3 m_k$ is even by the method of monodromy theory.