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Let $A=(a_{ij})_{n\times n}$ be an invertible matrix and$A^{-1}=(a^{ij})_{n\times n}$ be the inverse of $A$. In this paper, we considerthe generalized Liouville system: \label{abeq1} \Delta_g u_i+\sum_{j=1}^na_{ij}\rho_j(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}-1)=0\quad\text{in \,}M, where$0< h_j\in C^1(M)$ and $\rho_j\in \mathbb R^+$, and prove that, under theassumptions of $(H_1)$ and $(H_2)$\,(see Introduction), the Leray-Schauderdegree of \eqref{abeq1} is equal to \frac{(-\chi(M)+1)... (-\chi(M)+N)}{N!} if$\rho=(\rho_1,..., \rho_n)$ satisfies 8\pi N\sum_{i=1}^n\rho_i<\sum_{1\leqi,j\leq n}a_{ij}\rho_i\rho_j<8\pi(N+1)\sum_{i=1}^n\rho_i. Equation\eqref{abeq1} is a natural generalization of the classic Liouville equation andis the Euler-Lagrangian equation of Nonlinear function $\varPhi_\rho$:\varPhi_\rho(u)=1/2\int_M\sum_{1\leq i,j\leq n}a^{ij}\nabla_g u_i\cdot \nabla_gu_j+\sum_{i=1}^n\int_M\rho_iu_i -\sum_{i=1}^n\rho_i\log \int_M h_i e^{u_i}. TheLiouville system \eqref{abeq1} has arisen in many different research areas inmathematics and physics. Our counting formulas are the first result in degreetheory for Liouville systems.

A Topological Degree Counting for some Liouville Systems of Mean Field Equations