We consider a variant of the Lugiato-Lefever equation (LLE), which is a
nonlinear Schrödinger equation on a one-dimensional torus with forcing and
damping, to which we add a first-order derivative term with a potential
$\epsilon V(x)$. The potential breaks the translation invariance of LLE.
Depending on the existence of zeroes of the effective potential $V_\text{eff}$,
which is a suitably weighted and integrated version of $V$, we show that
stationary solutions from $\epsilon=0$ can be continued locally into the range
$\epsilon\not =0$. Moreover, the extremal points of the $\epsilon$-continued
solutions are located near zeros of $V_\text{eff}$. We therefore call this
phenomenon \emph{pinning} of stationary solutions. If we assume additionally
that the starting stationary solution at $\epsilon=0$ is spectrally stable with
the simple zero eigenvalue due to translation invariance being the only
eigenvalue on the imaginary axis, we can prove asymptotic stability or
instability of its $\epsilon$-continuation depending on the sign of
$V_\text{eff}'$ at the zero of $V_\text{eff}$ and the sign of $\epsilon$. The
variant of the LLE arises in the description of optical frequency combs in a
Kerr nonlinear ring-shaped microresonator which is pumped by two different
continuous monochromatic light sources of different frequencies and different
powers. Our analytical findings are illustrated by numerical simulations.