Consider for a closed Riemann surface $(\Sigma,j_{\Sigma})$, a compact Lie group $G$ and $P$ a principal $G$ bundle over $\Sigma$ and a connection 1-form $A$ on the principal bundle $P$ and $u:P \to M$ a $G$-equivariant map. Furthermore, let $(M,\omega)$ be a (symplectic) Hamiltonian $G$-Manifold with moment map $\mu:M\to \mathfrak{g}$ and an almost complex structure $J$ on $M$. Denote the field strength by $F_A$ and by $*$ the Hodge dual.
My question is: Does the following set of equations, called the symplectic vortex equations,
$$
\bar{\partial}_{J,A}u=\frac{1}{2}(d_Au+J\circ d_Au\circ j_{\Sigma})=0 \\
*F_A+\mu(u)=0
$$
always admit a solution?
I am aware the several cases have been already studied - e.g. for $M=\mathbb{C}^n$ and $G=U(1)$, but most of those cases are for abelian $G$. I am paricularly interested in the cases where $G=SO(3)$ and $M=S^2 \subset \mathfrak{so}(3)$ and the action is given by the adjoint action.
So far, apart from looking at the work in the Abelian cases, I figured out that an index argument could work out, however, my knowledge in algebraic topology is failing me here. I am thankful for any reference or hint, even for conjectures.