Give to Ursa Minor what is Ursa Minor's: an updated census of the RR Lyrae population in the Ursa Minor dwarf galaxy based on Gaia DR3

We use RR Lyrae stars identified by the Gaia third data release (DR3) to explore the outskirts of the Ursa Minor (UMi) dwarf spheroidal galaxy (dSph) and update the census of its variable star population. We adopted different tools based on the Gaia DR3 astrometric and photometric data (proper motions, Period-Wesenheit-Metallicity relations, spatial distribution, colour-magnitude diagram and stellar isochrone fitting) to discriminate between different types of variable stars, and to identify UMi members. We find a total of 129 RR Lyrae stars and Anomalous Cepheids (ACs) that belong to UMi. We report 47 new RR Lyrae stars (46 bona fide and 1 candidate) and 5 new ACs (4 bona fide and 1 candidate), including new possible members in the extreme periphery of the galaxy at a distance of $\sim$ 12 half-light radii. We reclassified 13 RR Lyrae stars identified by the Gaia DR3 Specific Object Study pipeline for Cepheids and RR Lyrae stars (SOS Cep$\&$RRL), using data from the literature and Gaia astrometry and photometry. Specifically, we assigned these 13 DR3 RR Lyrae stars to 10 Anomalous Cepheids and 3 double-mode RR Lyrae (RRd), respectively. From the average luminosity of the RR Lyrae stars we derive for UMi a distance modulus of $(m-M)_{0}=$ 19.23 $\pm$ 0.09 mag in excellent agreement with the literature. Finally, we investigated whether some of UMi's variable stars might be members of the ultra-faint stellar cluster Mu{\~n}oz~1, that lies at a projected distance of 45 arcmin from UMi's centre. Based on the properties of the variable stars (distances, colours and metallicities), we find unlikely that they belong to the cluster.