  In this paper we establish the existence of vortex solutions for a
Chern--Simons--Higgs model with gauge group $SU(N) \times U(1)$ and flavor
SU(N), these symmetries ensuring the existence of genuine non-Abelian vortices
through a color-flavor locking. Under a suitable ansatz we reduce the problem
to a $2\times 2$ system of nonlinear elliptic equations with exponential terms.
We study this system over the full plane and over a doubly periodic domain,
respectively. For the planar case we use a variational argument to establish
the existence result and derive the decay estimates of the solutions. Over the
doubly periodic domain we show that the system admits at least two
gauge-distinct solutions carrying the same physical energy by using a
constrained minimization approach and the mountain-pass theorem. In both cases
we get the quantized vortex magnetic fluxes and electric charges.
