  In this paper we give a geometric description of the foliation of a generic
real analytic family unfolding a real analytic vector field with a weak focus
at the origin, and show that two such families are orbitally analytically
equivalent if and only if the families of diffeomorphisms unfolding the
complexified Poincar\'e map of the singularities are conjugate. Moreover, by
shifting the leaves of the formal normal form in the blow-up (quasiconformal
surgery) by means of a fibered transformation along a convenient complex
cross-section, one constructs an abstract manifold of complex dimension 2
equipped with an elliptic holomorphic foliation whose monodromy map coincides
with a given family of admissible diffeomorphisms.
