  In this paper, we prove that there are no solutions for the curvature
equation \[ \Delta u+e^{u}=8\pi n\delta_{0}\text{ on }E_{\tau}, \quad
n\in\mathbb{N}, \] where $E_{\tau}$ is a flat rectangular torus and
$\delta_{0}$ is the Dirac measure at the lattice points. This confirms a
conjecture in \cite{CLW2} and also improves a result of Eremenko and Gabrielov
\cite{EG}. The nonexistence is a delicate problem because the equation always
has solutions if $8\pi n$ in the RHS is replaced by $2\pi \rho$ with
$0<\rho\notin 4\mathbb{N}$. Geometrically, our result implies that a
rectangular torus $E_{\tau}$ admits a metric with curvature $+1$ acquiring a
conic singularity at the lattice points with angle $2\pi\alpha$ if and only if
$\alpha$ is not an odd integer.
  Unexpectedly, our proof of the nonexistence result is to apply the spectral
theory of finite-gap potential, or equivalently the algebro-geometric solutions
of stationary KdV hierarchy equations. Indeed, our proof can also yield a sharp
nonexistence result for the curvature equation with singular sources at three
half periods and the lattice points.
