  Investigations of frequency combs in $\chi^{(3)}$ optical microresonators are
burgeoning nowadays. Changeover to $\chi^{(2)}$ 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
spatial walk-off. We predict walk-off controlled highly stable comb generation,
drastically different from that known in the $\chi^{(3)}$ case. This includes
the general notion of antiperiodic state, formation of coherent antiperiodic
steady states (solitons), where the FH and SH envelopes move with a common
velocity without shape changes, characterization of the family of antiperiodic
steady states, and the dependence of comb spectra on the pump power and the
group velocity difference.
