  Consider a smooth action $\mathbf G\times M \rightarrow M$ of a compact
connected Lie group $\mathbf G$ 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 $\mathbf G$ regarded as a group of
diffeomorphisms of $M$ (the existence of a free point implies that the action
of $\mathbf G$ is effective). Moreover, some examples of effective actions with
no free point where this result fails are exhibited.
