On reduction of singularities for vector felds in dimension 3

A classical result due to Seidenberg states that every singular holomorphic foliation on a complex surface can be turned into a foliation possessing only elementary singular points by means of a finite sequence of (one-point) blow-ups. Here we remind the reader that a singular point $p$ is said to be elementary if the foliation $\mathcal F$ in question has at least one eigenvalue different from zero at $p$. However, in dimension 3, the natural analogue of Seidenberg theorem no longer holds as shown by Sanz and Sancho.
More recently, two major works have been devoted to this problem. In [1], Cano, Roche and Spivakovsky have worked out a reduction procedure using (standard) blow-ups. The main disadvantage of their theorem lies, however, in the fact that some of their final models are quadratic and hence have all eigenvalues equal to zero. On the other hand, McQuillan and Panazzolo [2] have successfully used weighted blow-ups to obtain a satisfactory desingularization theorem in the category of stacks, rather than in usual complex manifolds.
A basic question is how far these theorems can be improved if we start with a complete vector field on a complex manifold of dimension 3, rather than with a general 1-dimensional holomorphic foliation. In this context of complete vector fields, we will prove a sharp desingularization theorem. Our proof of the mentioned result will naturally require us to revisit the works of Cano-Roche-Spivakovsky and of McQuillan-Panazzolo on general 1-dimensional foliations. In particular, by building on [1], our discussion will also shed some new light on the desingularization problem for general 1-dimensional foliation on complex manifolds of dimension 3.