Investigations of frequency combs in χ(2) optical microresonators are burgeoning nowadays. Changeover to χ(2) resonators promises further advances and brings new challenges. Here, the comb generation entails not only coupled first and second harmonics (FHs and SHs) and two dispersion coefficients but also a substantial difference in the group velocities - the temporal walk-off. We predict walk-off controlled highly stable comb generation, which is drastically different from that known in the χ(2) case. This includes the general notion of antiperiodic states; formation of localized coherent antiperiodic steady states (solitons), where the FH and SH envelopes move with a common velocity without shape changes; characterization of a new vast family of antiperiodic solitons; and the dependence of comb spectra on the pump power and the group velocity difference.
