In this paper, we study some aspects of the geometry of leaves of integrable distributions of lightlike hypersurfaces in indefinite Kenmotsu manifolds, tangent to the structure vector field. Theorems on parallel vector field, Killing distribution, geodesibility of lightlike hypersurfaces are obtained. Some characterization Theorems of leaves of integrable distributions are given. We prove that there exists a distribution, subset of the screen distribution, in which, under the integrability condition, any totally contact umbilical leaf is an extrinsic sphere.