  We report the discovery of a new group of double-periodic stars in the OGLE
Galactic bulge photometry. In 38 stars identified as fundamental mode RR~Lyrae
and 4 classified as the first-overtone RR~Lyrae, we detected additional shorter
periodicity. Periods of the dominant variability in the newly discovered group
are $0.28<P_{\rm D}<0.41$\,days. Period ratios $(0.68 - 0.72)$ are smaller than
the period ratios of the Galactic bulge RRd stars. The typical amplitude ratio
(of the additional to the dominant periodicity) is 20\,\% for the stars
identified as fundamental mode RR~Lyrae and 50\,\% for stars classified as the
first-overtone RR~Lyrae. Ten stars from our sample exhibit equidistant peaks in
the frequency spectrum, that suggest the Blazhko-type modulation of the main
pulsation frequency and/or the additional periodicity. The Fourier coefficients
$R_{\mathrm{21}}$ and $R_{\mathrm{31}}$ are one of the lowest among fundamental
mode RR Lyrae stars, but among the highest for the first-overtone pulsators.
For the phase Fourier coefficients $\varphi_{\mathrm{21}}$ and
$\varphi_{\mathrm{31}}$, our stars lie in between RRab and RRc stars. Discussed
stars were compared with the radial, linear pulsation models. Their position in
the Petersen diagram cannot be reproduced assuming that two radial modes are
excited and their physical parameters are similar to that characteristic for
RR~Lyrae stars. The non-radial mode scenario also faces difficulties. We
conclude that the dominant variability is most likely due to pulsation in the
radial fundamental mode including stars classified as the first overtone mode
pulsators. At this point, we cannot explain the nature of the additional
periodicity. Even more, classification of the stars as RR~Lyrae should be
treated as tentative.
