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 (free point). Then we prove that there exist two complete vector field X, X 1 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).Besides, some examples of effective actions with no free point where this result fails are exhibited.
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