  In the present paper, we consider a $\left(1 + 1\right)$-dimensional gauge
model consisting of two complex scalar fields interacting with each other
through an Abelian gauge field. When the model's gauge coupling constants are
set equal to zero, the model possesses non-gauged Q-ball and kink solutions
that do not interact with each other. It is shown that at nonzero gauge
coupling constants, the model possesses the soliton solution describing the
system consisting of interacting Q-ball and kink components. The kink and
Q-ball components of the kink-Q-ball system have opposite electric charges, so
the total electric charge of the kink-Q-ball system vanishes. Properties of the
kink-Q-ball system are researched analytically and numerically. In particular,
it was found that the kink-Q-ball system possesses a nonzero electric field and
is unstable with respect to small perturbations of fields.
