  We disentangle two counter-rotating stellar components in NGC 4191 and
characterize their physical properties (kinematics, morphology, age,
metallicity, and abundance ratio). We performed a spectroscopic decomposition
on integral field data to separate the contribution of two stellar components
to the observed galaxy spectrum across the field of view. We also performed a
photometric decomposition, modelling the galaxy with a S\'ersic bulge and two
exponential disks of different scale length, with the aim of associating these
structural components with the kinematic components. We measured the equivalent
width of the absorption line indices on the best fit that represent the
kinematic components and compared our measurements to the predictions of
stellar population models. We have evidence that the line-of-sight velocity
distributions (LOSVDs) are consistent with the presence of two distinct
kinematic components. The combined information of the intensity of the LOSVDs
and photometry allows us to associate the S\'ersic bulge and the outer disk
with the main kinematic component, and the inner disk with the secondary
kinematic component. The two kinematic stellar components counter-rotate with
respect to each other. The main component is the most luminous and massive, and
it rotates slower than the secondary component, which rotates along the same
direction as the ionized gas. We also found that the two kinematic components
have the same solar metallicity and sub-solar abundance ratio, without the
presence of significant radial gradients. On the other hand, their ages show
strong negative gradients and the possible indication that the secondary
component is the youngest. We interpret our results in light of recent
cosmological simulations and suggest gas accretion along two filaments as the
formation mechanism of the stellar counter-rotating components in NGC 4191
(Abridged).
