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Revisiting the variable star population in NGC~6229 and the structure of the Horizontal Branch

We report an analysis of new $V$ and $I$ CCD time-series photometry of the distant globular cluster NGC 6229. The principal aims were to explore the field of the cluster in search of new variables, and to Fourier decompose the RR Lyrae light curves in pursuit of physical parameters.We found 25 new variables: 10 RRab, 5 RRc, 6 SR, 1 CW, 1 SX Phe, and two that we were unable to classify. Secular period changes were detected and measured in some favourable cases. The classifications of some of the known variables were rectified. The Fourier decomposition of RRab and RRc light curves was used to independently estimate the mean cluster value of [Fe/H] and distance. From the RRab stars we found [Fe/H]$_{\rm UVES}$=$-1.31 \pm 0.01{\rm(statistical)} \pm 0.12{\rm(systematic)}$ ([Fe/H]$_{\rm ZW}=-1.42$),and a distance of $30.0\pm 1.5$ kpc, and from the RRc stars we found [Fe/H]$_{\rm UVES}$=$-1.29\pm 0.12$ and a distance of $30.7\pm 1.1$ kpc, respectively. Absolute magnitudes, radii and masses are also reported for individual RR Lyrae stars. Also discussed are the independent estimates of the cluster distance from the tip of the RGB, 34.9$\pm$2.4 kpc and from the P-L relation of SX Phe stars, 28.9$\pm$2.2 kpc. The distribution of RR Lyrae stars in the horizontal branch shows a clear empirical border between stable fundamental and first overtone pulsators which has been noted in several other clusters; we interpret it as the red edge of the first overtone instability strip.