On computing canonical forms using flows

Let G be a Lie group acting on a vector space . I say that a vector field is orbital if for all p in , u(p) is tangent to the orbit of p at p. Now let the vector space and the vector space tangent to G at the identity have inner products. In this setting I define a simple map (which I call quasi-projection) which transforms any vector field on into an orbital one. I use the quasi-projection map to define flows which compute canonical forms.