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 Sersic 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 Sersic 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).
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