Let a torus act smoothly on a manifold. Assume that the manifold is connected and the action is faithful. Then every orbit-preserving equivariant diffeomorphism is obtained from an invariant smooth map from the manifold to the torus by acting by the values of this function.

This theorem - in a more general setup of torus actions on orbifolds - appeared in Haefliger and Salem's 1991 paper
"Actions of tori on orbifolds". The crucial step of the proof was attributed to Gerald Schwarz.

Together with Gerald Schwarz, we close a gap in this proof.