Exact counting of spherical metrics with one conical singularity on rectangular tori

We prove that for every integer $n\geq 2$ and $8π(n-1)<ρ<8πn$, the singular Liouville equation $Δu+\e^u=ρδ_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Li