  Consider a smooth effective action of a torus $\mathbb{T}^n$ on a connected
$C^{\infty}$-manifold $M$ of dimension $m$. Then $n\leq m$. 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
${\operatorname{Diff}}(M)$.
