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-François Mattei (Invent Math 103(2):297–325, 1991 , Theorem 3.2.1) about the existence of local slices for a (non-compact) Lie group action.