  Vortices produce locally concentrated field configurations and are solutions
to the nonlinear partial differential equations systems of complicated
structures. In this paper, we establish the existence and uniqueness for
solutions of the gauged non-Abelian vortices in a coupled 4D-2D quantum field
theory by researching the nonlinear elliptic equations systems with exponential
terms in $\mathbb{R}^{2}$ using the calculus of variations. In addition, we
obtain the asymptotic behavior of the solutions at infinity and the quantized
integrals in $\mathbb{R}^{2}$.
