Foliations with transversal quaternionic structures

We consider manifolds equipped with a foliation T of codimension 4q, and an almost quaternionic structure g on the transversal bundle of P'. After discussing conditions of projectability and integrability of Q, we study the transversal twistor space Zf which, by definition, consists of the Q-compatible almost complex structures. We show that Zf can be endowed with a liftedjbliation F and two natural almost complex structures J\, Ji on the transversal bundle of y'. We establish the conditions which ensure the projectability of J₁ and J2, and the integrability of J₁ (J₂ is never integrable).