In a recent paper a class of complex, compact and non-Kählerian manifolds was constructed by S. López de Medrano and A. Verjowsky. This class contains as particular cases Calabi-Eckmann manifolds, almost all Hopf manifolds and many of the examples given previously by J.-J. Loeb and M. Nicolau. In this paper we show that these manifolds are endowed with a natural non-singular vector field which is transversely Kählerian, and that analytic subsets of appropriate dimensions are tangent to this vector field. This permits to give a precise description of these sets in the generic case. In the proof, an important role is played by some complex abelian groups which are biholomorphic to big domains in these manifolds.