Sharp criteria for the existence of solutions to curvature equations with two singular sources on rectangular torus

We study the following curvature equation with two singular sources on a flat torus $$ Δu+e^u=8π(δ_p+δ_{-p})\quad\text{on } E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ), $$ where $δ_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In this