By this answer I know that
The existence of a nontrivial Killing vector field $\xi$ on a compact Riemannian manifold $M$ is equivalent to the existence of a nontrivial $\Bbb S^1$-action on $M$.
where $\xi$ is called Killing if $\mathcal{L}_\xi g=0$. There is a generalization of Killing vector fields as follows:
Definition: A vector field $\xi$ is called $2$-Killing if $\mathcal{L}_\xi\mathcal{L}_\xi g=0$.
Question: What is the relation between "existence of a nontrivial $2$-Killing vector field on a compact Riemannian manifold" and "existence of a nontrivial $\Bbb T^k$-action"?
I know that if $\xi$ is Killing then it is $2$-Killing. So $\Bbb S^1\subset $ isometries generated by $2$-Killing vector fields?