  The formation scenario of extended counter-rotating stellar disks in galaxies
is still debated. In this paper, we study the S0 galaxy IC 719 known to host
two large-scale counter-rotating stellar disks in order to investigate their
formation mechanism. We exploit the large field of view and wavelength coverage
of the Multi Unit Spectroscopic Explorer (MUSE) spectrograph to derive
two-dimensional (2D) maps of the various properties of the counter-rotating
stellar disks, such as age, metallicity, kinematics, spatial distribution, the
kinematical and chemical properties of the ionized gas, and the dust map. Due
to the large wavelength range, and in particular to the presence of the Calcium
Triplet 8498, 8542, 8662\AA (CaT hereafter), the spectroscopic analysis allows
us to separate the two stellar components in great detail. This permits precise
measurement of both the velocity and velocity dispersion of the two components
as well as their spatial distribution. We derived a 2D map of the age and
metallicity of the two stellar components as well as the star formation rate
and gas-phase metallicity from the ionized gas emission maps. The main stellar
disk of the galaxy is kinematically hotter, older, thicker and with larger
scale-length than the secondary disk. There is no doubt that the latter is
strongly linked to the ionized gas component: they have the same kinematics and
similar vertical and radial spatial distribution. This result is in favor of a
gas accretion scenario over a binary merger scenario to explain the origin of
counter-rotation in IC 719. One source of gas that may have contributed to the
accretion process is the cloud that surrounds IC 719.
