  We present, for the first time, spatially resolved spectroscopy for the
entire Hickson Compact Group 31 obtained with the MUSE instrument at the
VLT,and an in-depth analysis of this compact group. To obtain a complete
understanding of the system, we derived radial velocity and dispersion velocity
maps, maps of the ionization mechanism of the system, chemical abundances and
their distribution over the whole system, star formation rates and ages of the
different star-forming regions, and the spatial distribution of the Wolf-Rayet
stellar population. We also reconstructed the star formation history of the
galaxies HCG 31 A, C, B and F, measured the emission-line fluxes, and performed
a stellar population synthesis. Our main findings are: (i) that there is
clearly disturbed kinematics due to the merger event that the system is
experiencing; (ii) that the ionization is produced exclusively via star
formation except for the nucleus of the galaxy HCG 31 A, where there is a small
contribution of shocks; (iii) that there is low oxygen abundance distributed
homogeneously through the system; (iv) that there is a prominent population of
carbon Wolf-Rayet stars in the central zone of the group; and (v) that there
are clear evidences of the tidal origin of the galaxies HCG 31 E, HCG 31 H, and
HCG 31 F because they show quite high oxygen abundances for their stellar mass.
All these findings are clear evidence that HCG 31 is currently in an early
merging phase and manifesting a starburst in its central region.
