We will show that the right set up for questions concerning the
existence of foliated generic function is that of Levi flat CR
manifolds which embed in projective space. More generally, the notion of
Lefschetz pencil structure (a kind of CR morse function) will be
defined. Such structures will be shown to exist for projective CR
manifolds. The particular case of taut foliations in 3-dimensional
manifolds will be analyzed. If time allows, the relation between the
notion of strict C-convexity and Lefschetz pencil structures of the
simplest kind will we sketched.