Classical Cepheids (DCEPs) are important astrophysical objects not only as standard candles in the determination of the cosmic distance ladder, but also as a testbed for the stellar evolution theory, thanks to the strict connection between their pulsation [period(s), amplitudes] and stellar (luminosity, mass, effective temperature, metallicity) parameters. We aim at unveiling the nature of the Galactic DCEP V363 Cas and other DCEPs showing cosmic abundances of lithium in their atmospheres. We have collected three epochs high-resolution spectroscopy for V363 Cas with HARPS-N@TNG. Accurate stellar parameters: effective temperatures, gravities, micro-turbulences, radial velocities, and metal abundances were measured for this star. We detected a lithium abundance of A(Li)=2.86+-0.10 dex, along with iron, carbon and oxygen abundances of [Fe/H]=-0.30+-0.12 dex, [C/H]=-0.06+-0.15 dex and [O/H]=0.00+-0.12 dex. V363 Cas is the fifth among the Milky Way DCEPs to exhibit a Li-rich feature. An analysis of historical time-series spanning a hundred year interval shows that the period of V363 Cas is increasing, with a sharp acceleration after HJD=2453000. This is a clear hint of first crossing of the instability strip. Our results favour the scenario in which the five Galactic Li-rich DCEPs are first-crossing the instability strip having had slowly-rotating progenitors during their main sequence phase.
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