  We present a series of existence theorems for multiple vortex solutions in
the Gudnason model of the ${\cal N}=2$ supersymmetric field theory where
non-Abelian gauge fields are governed by the pure Chern--Simons dynamics at
dual levels and realized as the solutions of a system of elliptic equations
with exponential nonlinearity over two-dimensional domains. In the full plane
situation, our method utilizes a minimization approach, and in the doubly
periodic situation, we employ an-inequality constrained minimization approach.
In the latter case, we also obtain sufficient conditions under which we show
that there exist at least two gauge-distinct solutions for any prescribed
distribution of vortices. In other words, there are distinct solutions with
identical vortex distribution, energy, and electric and magnetic charges.
