Smooth toric actions are described by a single vector field

Consider a smooth effective action of a torus ${\mathbb{T}}^n$ on a connected $C^{\mathrm{\infty }}$-manifold $M$ of dimension $m$. Then $n\le m$. In this work we show that if $n<m$, then there exist a complete vector field $X$ on $M$ such that the automorphism group of $X$ equals ${\mathbb{T}}^n\otimes \mathbb{R}$, where the factor $\mathbb{R}$ comes from the flow of $X$ and ${\mathbb{T}}^n$ is regarded as a subgroup of the full group of diffeomorphisms of $\mathrm{Diff}(M)$.