Smooth actions of connected compact Lie groups with a free point are determined by two vector fields

Consider a smooth action G×M→M of a compact connected Lie group G on a connected manifold M. Assume the existence of a point of M whose isotropy group has a single element (a free point). Then we prove that there exist two complete vector field X,X1 such that their group of automorphisms equals G regarded as a group of diffeomorphisms of M (the existence of a free point implies that the action of G is effective). Moreover, some examples of effective actions with no free point where this result fails are exhibited.