  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 $\mathcal{T}M$ 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 $\mathcal{T}M$ which are integrable. We also find
geometric properties of both leaves of Liouville distribution and the vertical
distribution in our context.
