Foliations with symplectic leaves and their constructions are drawing attentions not only as a foliated symplectic geometry but also as important examples of Poisson geometry, because they are nothing but regular Poisson structures. On the other hand, as non-trivial examples not many are known at the present, for instance, on the 7-sphere the existence of codimension 1 symplectic foliations is still an open problem.
A method of construction on $S^5$ is explained in terms of the contact geometry and foliations of exact symplectic open books. We need to destroy the symplectic convexity of Milnor fibers of simple elliptic or cusp singularities. This formulation allowed A. Mori to rpoduce an interesting family of examples on $S^1 \times S^4$.
Foliated Lefschetz fibration is another technology to produce examples. If the time allows, we apply this technology to our examples and approach to understanding closed symplectic 4-manifolds obtained by our method. This part is a joint work (in progress) with N. Kasuya, H. Kodama, and A.Mori.