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We consider Kerr frequency combs in a dual-pumped microresonator astime-periodic and spatially $2\pi$-periodic traveling wave solutions of avariant of the Lugiato-Lefever equation, which is a damped, detuned and drivennonlinear Schr\"odinger equation given by $\mathrm{i}a_\tau=(\zeta-\mathrm{i})a - d a_{xx}-|a|^2a+\mathrm{i}f_0+\mathrm{i}f_1\mathrm{e}^{\mathrm{i}(k_1 x-\nu_1\tau)}$. The main new feature of the problem is the specific form of the sourceterm $f_0+f_1\mathrm{e}^{\mathrm{i}(k_1 x-\nu_1 \tau)}$ which describes thesimultaneous pumping of two different modes with mode indices $k_0=0$ and$k_1\in \mathbb{N}$. We prove existence and uniqueness theorems for thesetraveling waves based on a-priori bounds and fixed point theorems. Moreover, byusing the implicit function theorem and bifurcation theory, we show hownon-degenerate solutions from the $1$-mode case, i.e. $f_1=0$, can be continuedinto 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 numericalsimulations.

Global continua of solutions to the Lugiato-Lefever model for frequency combs obtained by two-mode pumping