  Let $(M,g)$ be a compact Riemann surface with no boundary and $u=(u_1,u_2)$
be a solution of the following singular Liouville system: $$\Delta_g
u_i+\sum_{j=1}^2 a_{ij}\rho_j(\frac{h_je^{u_j}}{\int_M
h_je^{u_j}dV_g}-1)=\sum_{l=1}^{N}4\pi\gamma_l(\delta_{p_l}-1), $$ where
$h_1,h_2$ are positive smooth functions, $p_1,\cdots,p_N$ are distinct points
on $M$, $\delta_{p_l}$ are Dirac masses, $\rho=(\rho_1,\rho_2)(\rho_i\geq 0)$
and $(\gamma_1,\cdots,\gamma_N)(\gamma_l > -1)$ are constant vectors. In the
previous work, we derive a degree counting formula for the singular Liouville
system when $A$ satisfies standard assumptions. In this article, we establish a
more general degree counting formula for 2$\times$2 singular Liouville system
when the coefficient matrix $A$ is non-symmetric and non-invertible. Finally,
the existence of solution can be proved by the degree counting formula which
depends only on the topology of the domain and the location of $\rho$.
