  Over the recent years, additional low-amplitude non-radial modes were
detected in many of the first-overtone RR Lyrae stars. These non-radial modes
form a characteristic period ratio with the dominant first-overtone mode of
around 0.61. The incidence rate of this phenomenon changes from population to
population. It is also strongly dependent on the quality of the analyzed data.
Current models explaining these additional signals involve non-radial modes of
degrees 8 and 9. Using synthetic horizontal branch populations, we investigate
the incidence rate of first-overtone RR Lyrae stars with non-radial modes
depending on the population properties, i.e., ages and metallicities. We
compare our results with the observed results for globular clusters and the
numerous collection of field first-overtone RR Lyrae stars to test the
predictions of the models. We used synthetic horizontal branches combined with
pulsation models to predict how the incidence rate would depend on the age and
metallicity of the population. To test whether the results based on synthetic
horizontal branches are realistic, we compared them to incidence rates observed
by TESS in first-overtone field RR Lyrae stars, using photometric metallicity
values from a newly established calibration for TESS. The analysis of synthetic
horizontal branches showed that the incidence rate decreases with decreasing
metallicity. We inferred photometric metallicity for RR Lyrae stars observed by
TESS and showed that the theoretical predictions are in agreement with the
observations. Using the same method, we also conclude that the metallicity
distribution of RR Lyrae stars showing an additional mode with a period-ratio
around $0.68$ appears to be different from that of both all first-overtone
stars and those showing additional non-radial modes.
