  We consider Kerr frequency combs in a dual-pumped microresonator as
time-periodic and spatially $2\pi$-periodic traveling wave solutions of a
variant of the Lugiato-Lefever equation, which is a damped, detuned and driven
nonlinear Schr\"odinger equation given by $\mathrm{i}a_\tau
=(\zeta-\mathrm{i})a - d a_{x
x}-|a|^2a+\mathrm{i}f_0+\mathrm{i}f_1\mathrm{e}^{\mathrm{i}(k_1 x-\nu_1
\tau)}$. The main new feature of the problem is the specific form of the source
term $f_0+f_1\mathrm{e}^{\mathrm{i}(k_1 x-\nu_1 \tau)}$ which describes the
simultaneous pumping of two different modes with mode indices $k_0=0$ and
$k_1\in \mathbb{N}$. We prove existence and uniqueness theorems for these
traveling waves based on a-priori bounds and fixed point theorems. Moreover, by
using the implicit function theorem and bifurcation theory, we show how
non-degenerate solutions from the $1$-mode case, i.e. $f_1=0$, can be continued
into the range $f_1\not =0$. Our analytical findings apply both for anomalous
($d>0$) and normal ($d<0$) dispersion, and they are illustrated by numerical
simulations.
