  We present the results of integral-field spectroscopic observations of the
two disk galaxies NGC 3593 and NGC 4550 obtained with VIMOS/VLT. Both galaxies
are known to host 2 counter-rotating stellar disks, with the ionized gas
co-rotating with one of them. We measured in each galaxy the ionized gas
kinematics and metallicity, and the surface brightness, kinematics, mass
surface density, and the stellar populations of the 2 stellar components to
constrain the formation scenario of these peculiar galaxies. We applied a novel
spectroscopic decomposition technique to both galaxies, to separate the
relative contribution of the 2 counter-rotating stellar and one ionized-gas
components to the observed spectrum. We measured the kinematics and the line
strengths of the Lick indices of the 2 counter-rotating stellar components. We
modeled the data of each stellar component with single stellar population
models that account for the alpha/Fe overabundance. In both galaxies we
successfully separated the main from the secondary stellar component that is
less massive and rotates in the same direction of the ionized-gas component.
The 2 stellar components have exponential surface-brightness profiles. In both
galaxies, the two counter-rotating stellar components have different stellar
populations: the secondary stellar disk is younger, more metal poor, and more
alpha-enhanced than the main galaxy stellar disk. Our findings rule out an
internal origin of the secondary stellar component and favor a scenario where
it formed from gas accreted on retrograde orbits from the environment fueling
an in situ outside-in rapid star formation. The event occurred ~ 2 Gyr ago in
NGC 3593, and ~ 7 Gyr ago in NGC 4550. The binary galaxy merger scenario cannot
be ruled out, and a larger sample is required to statistically determine which
is the most efficient mechanism to build counter-rotating stellar disks
(abridged).
