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The realization of single-mode generation in a microwave oscillator is considered for a relativistic resonant traveling-wave tube (RTWT) with spiral Bragg reflectors at the ends of a multimode electrodynamic system. An oversized azimuthally symmetric slow-wave structure was chosen so that only two field structures are synchronous with a 1-MeV electron beam. Experiments using different means that allow us to separate these field structures are described. As a result, two radiation patterns were observed separately, which are a narrow Gaussian-like wave beam with a power of up to 1.5 GW and an efficiency of about 20% in 40-ns pulses and radiation corresponding to the E01 mode with four times weaker power. The dependences of RTWT operation on the parameters of the electron beam, the length of interaction space, reflection coefficients, and the amplitude of guide axial magnetic field are investigated in detail.