Symmetry of solutions of a mean field equation on flat tori

We study symmetry of solutions of the mean field equation $$\mathrm{\Delta }u+\rho (\frac{K{e}^u}{{\int }_{T_{ϵ}}K{e}^u}-\frac{1}{\mathrm{\mid }{T}_{ϵ}\mathrm{\mid }})=0$$ on the flat torus $T_\epsilon=[-\frac{1}{2\epsilon}, \frac{1}{2\epsilon}] \times [-\frac{1}{2}, \frac{1}{2}]$with$0<\epsilon \leq 1$,where$K\in C^2({T}\epsilon)$isapositivefunctionwith$-\Delta \ln K \leq \frac{\rho}{|T\epsilon|}$and$\rho \leq 8\pi$\mathrm{.}Weprovethatif$(x_0,y_0)$isacriticalpointofthefunction$u+ln(K)$,then$u$ is evenly symmetric about the lines $x=x_0$ and $y=y_0$, provided $K$ is evenly symmetric about these lines. In particular we show that all solutions are one-dimensional if $K\equiv 1$ and $\rho \le 8\pi$. The results are sharp and answer a conjecture of Lin and Lucia affirmatively. We also prove some symmetry results for mean field equations on annulus.