  We consider a Toda system of Liouville equations defined on a compact surface
which arises as a model for non-abelian Chern-Simons vortices. For the first
time the range of parameters $\rho_1 \in (4k\pi , 4(k+1)\pi)$, $k \in
\mathbb{N}$, $\rho_2 \in (4\pi, 8\pi )$ is studied with a variational approach
on surfaces with arbitrary genus. We provide a general existence result by
means of a new improved Moser-Trudinger type inequality and introducing a
topological join construction in order to describe the interaction of the two
components.
