The present paper unifies some aspects concerning the vertical Liouville distributions on the tangent (cotangent) bundle of a Finsler (Cartan) space in the context of generalized geometry. More exactly, we consider the big-tangent manifold $\mathcal TM$ associated to a Finsler space $(M,F)$ and of its $\mathcal L$-dual which is a Cartan space $(M,K)$ and we define three Liouville distributions on $\mathcal TM$ which are integrable. We also find geometric properties of both leaves of Liouville distribution and the vertical distribution in our context.