  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 [$\alpha$/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
[$\alpha$/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.
