We analyzed photometry for the up-to-date collection of the first-overtone RR Lyrae stars (RRc; 11415 stars) and double-mode RR Lyrae stars (RRd; 148 stars) towards the Galactic bulge from the Optical Gravitational Lensing Experiment. We analyzed frequency spectra of these stars in search for additional, low-amplitude signals, beyond the radial modes. We focused on stars from two groups: $RR_{0.61}$ and $RR_{0.68}$. In the first group, additional low-amplitude signals have periods shorter than the first-overtone period; period ratios fall in the 0.60-0.64 range. In the second group, additional low-amplitude signals have periods longer than the first-overtone period; period ratios tightly cluster around 0.68. Altogether we have detected 960 and 147 RR Lyrae stars that belong to $RR_{0.61}$ and $RR_{0.68}$ groups, respectively, which yield the incidence rates of 8.3 and 1.3 per cent of the considered sample. We discuss statistical properties of RR Lyrae stars with additional periodicities. For $RR_{0.61}$ group we provide strong arguments that additional periodicities are connected to non-radial pulsation modes of degrees $ell=8$ and $ell=9$, as proposed by Dziembowski. We have also detected two double-periodic variables, with two close periodicities, similar to RR Lyrae variable V37 in NGC 6362. Properties of these peculiar variables, which may form a new group of double-mode pulsators, are discussed.
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