Orbits of Families of Vector Fields and Integrability of Systems with Singularities

This defines an equivalence relation on M. The equivalence classes are called the orbits of D. Let S be an orbit of D. For each me S and each finite sequence £ = (X , . . . , X) of elements of D, let F^m denote the map {tu...,ta^Xll(XfjL---Xl{m)---)). It is clear that F^m is a C 00 mapping from an open subset U of R into M. Moreover the range of F^m is a subset of 5. We topologize S by the strongest topology for which all the maps F^m are continuous. THEOREM 1. S is a connected C submanifold ofM.