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Massive ETGs are thought to form through a two-phase process. At early times, an intense and fast starburst forms blue and disk-dominated galaxies. After quenching, the remaining structures become red, compact and massive, i.e., 'red nuggets'. Then, a time-extended second phase which is dominated by mergers, causes structural evolution and size growth. Given the stochastic nature of mergers, a small fraction of red nuggets survives, without any interaction, massive and compact until today: relic galaxies. Since this fraction depends on the processes dominating the size growth, counting relics at low-z is a valuable way to disentangle between different galaxy evolution models. In this paper, we introduce the INvestigating Stellar Population In RElics (INSPIRE) Project, that aims at spectroscopically confirming and fully characterizing a large number of relics at 0.1<z<0.5. We focus here on the first results based on a pilot program targeting three systems, representative of the whole sample. For these, we extract 1D optical spectra over an aperture comprising ~30 % of the galaxies light, and obtain line-of-sight integrated stellar velocity and velocity dispersion. We also infer the stellar [$α$/Fe] abundance from line-index measurements and mass-weighted age and metallicity from full-spectral fitting with single stellar population models. Two galaxies have large integrated stellar velocity dispersion values, confirming their massive nature. They are populated by stars with super-solar metallicity and [$α$/Fe]. Both objects have formed >80 % of their stellar mass within a short (0.5 - 1.0 Gyrs) initial star formation episode occurred only ~1 Gyr after the Big Bang. The third galaxy has a more extended star formation history and a lower velocity dispersion. Thus we confirm two out of three candidates as relics.