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It is shown that the known conditions for cyclotron resonances are strictly valid only under autoresonance conditions or in the nonrelativistic case. In other cases, it is necessary to use the conditions written out in the work. The results of the main features of the dynamics of charged particles under new resonant conditions are presented. The main feature of particle dynamics is their stepwise nature. The temporal width of these steps was determined, as well as the moments of particle hopping from one step to another. The high sensitivity of the particle dynamics to small changes in the wave parameters is demonstrated. Conditions are found for practically unlimited acceleration of electrons by a transverse electromagnetic wave in a vacuum without a magnetic field. It is shown that such an acceleration can be carried out by a field whose strength parameter is less than unity. The results of acceleration of particles by the field of a Gaussian beam in vacuum are presented.