  The study of microwave and radiofrequency multi-photon transitions in
continuous wave (CW) EPR spectroscopy is extended to a Rabi resonance
condition, when the radio frequency of the magnetic-field modulation matches
the Rabi frequency of a spin system in the microwave field. Using the
non-secular perturbation theory based on the Bogoliubov averaging method, the
analytical description of the response of the spin system is derived for all
modulation frequency harmonics. When the modulation frequency exceeds the EPR
linewidth, multi-photon transitions result in sidebands in absorption EPR
spectra measured with phase-sensitive detection at any harmonic. The saturation
of different-order multi-photon transitions is shown to be significantly
different and to be sensitive to the Rabi resonance. The noticeable frequency
shifts of sidebands are found to be the signatures of this resonance. The
inversion of two-photon lines in some spectral intervals of the out-of-phase
first-harmonic signal is predicted under passage through the Rabi resonance.
The inversion indicates the transition from absorption to stimulated emission
or vice versa, depending on the sideband. The manifestation of the primary and
secondary Rabi resonance is also demonstrated in time-resolved steady-state EPR
signals formed by all harmonics of the modulation frequency. Our results
provide a theoretical framework for future developments in multi-photon CW EPR
spectroscopy, which can be useful for samples with long spin relaxation times
and extremely narrow EPR lines
