Generalized Flow-Box property for singular foliations

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.