Smooth torus actions are described by a single vector field.

Consider a smooth effective action of a torus Tn on a connected C∞-manifold M. Assume that M is not a torus endowed with the natural action. Then we prove that there exists a complete vector field X on M such that the automorphism group of X equals Tn×R, where the factor R comes from the flow of X and Tn is regarded as a subgroup of Diff(M). Thus one may reconstruct the whole action of Tn from a single vector field. [ABSTRACT FROM AUTHOR]