  In this paper we establish a multiplicity result concerning the existence of
doubly periodic solutions for a $2\times2$ nonlinear elliptic system arising in
the study of self-dual non-Abelian Chern--Simons vortices. We show that the
given system admits at least two solutions when the Chern--Simons coupling
parameter $\kappa>0$ is sufficiently small; while no solutions exist for
$\kappa>0$ sufficiently large. As in [36] we use a variational formulation of
the problem. Thus, we obtain a first solution via a (local) minimization method
and show that it is asymptotically gauge-equivalent to the (broken) principal
embedding vacuum of the system, as $\kappa\to 0$. Then we obtain the second
solution by a min-max procedure of "mountain pass" type.
