After a brief review on the basic notions and the principal results concerning
the Jacobi manifolds, the relationship between homogeneous Poisson manifolds and
conformal Jacobi manifolds, and also the compatible Jacobi manifolds, we give a
generalization of some of these results needed for the contents of this paper. We
introduce the notion of Jacobi-Nijenhuis structure and we study the relation between
Jacobi-Nijenhuis manifolds and homogeneous Poisson-Nijenhuis manifolds.
We present a local classification of homogeneous Poisson-Nijenhuis manifolds and
we establish some local models of Jacobi-Nijenhuis manifolds
