  We study a recently developed product Abelian gauge field theory by Tong and
Wong hosting magnetic impurities. We first obtain a necessary and sufficient
condition for the existence of a unique solution realizing such impurities in
the form of multiple vortices. We next reformulate the theory into an extended
model that allows the coexistence of vortices and anti-vortices. The two
Abelian gauge fields in the model induce two species of magnetic vortex-lines
resulting from $N_s$ vortices and $P_s$ anti-vortices ($s=1,2$) realized as the
zeros and poles of two complex-valued Higgs fields, respectively. An existence
theorem is established for the governing equations over a compact Riemann
surface $S$ which states that a solution with prescribed $N_1, N_2$ vortices
and $P_1,P_2$ anti-vortices of two designated species exists if and only if the
inequalities \[ \left|N_1+N_2-(P_1+P_2)\right|<\frac{|S|}{\pi},\quad
\left|N_1+2N_2-(P_1+2P_2)\right|<\frac{|S|}{\pi}, \] hold simultaneously, which
give bounds for the `differences' of the vortex and anti-vortex numbers in
terms of the total surface area of $S$. The minimum energy of these solutions
is shown to assume the explicit value \[ E= 4\pi (N_1+N_2+P_1+P_2), \] given in
terms of several topological invariants, measuring the total tension of the
vortex-lines.
