  We introduce a notion of generalized Flow-Box property valid for general
singular distributions and sub-varieties (based on a dynamical interpretation).
Just as in the usual Flow-Box Theorem, we characterize geometrical and
algebraic conditions of (quasi) transversality in order for an analytic
sub-variety $X$ (not necessarily regular) to be a section of a line foliation.
We also discuss the case of more general foliations.
  This study is originally motivated by a question of Jean-Francois Mattei
(concerning the strengthening of a Theorem of Mattei) about the existence of
local slices for a (non-compact) Lie group action.
