Mean field equations on a closed Riemannian surface with the action of an isometric group

Let $(Sigma,g)$ be a closed Riemannian surface, $textbf{G}={sigma_1,cdots,sigma_N}$ be an isometric group acting on it. Denote a positive integer $ell=inf_{xinSigma}I(x)$, where $I(x)$ is the number of all distinct points of the set ${sigma_1(x),cdots,sigma_N(x)}$. A sufficient condition for existence of solutions to the mean field equation $$Delta_g u=8piellleft(frac{he^u}{int_Sigma he^udv_g}-frac{1}{{rm Vol}_g(Sigma)}right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $ell=1$ or equivalently $textbf{G}={Id}$, where $Id$ is the identity map.