Consider $(M^{n},g)$ to be a Riemannian manifold and suppose that $X$ is a smooth non-trivial Killing vector field on $M$. Away from the zeros of $X$ we have a natural distribution $D$ of $(n-1)$-planes defined so that $D\_p$ is orthogonal to $X\_p$. If the distribution $D$ is (completely) integrable then it is straightforward to verify that the one form $\omega$ defined by
$$\omega(\cdot )=\frac{1}{g(X,X)} g(X, \cdot).$$
is closed (away from $\lbrace X=0\rbrace$). Moreover, the converse also holds.


Examples in $\mathbb{R}^n$ with the euclidean metric include the the translations along the $x\_i$-axis, $T\_i$ and rotations around the $x\_i$-axis, $R\_i$. The Killing fields $T\_i+R\_i$ are non-examples.


My question is whether this concept already has a name and where it might appear in the literature.



Special Killing Vector Fields
I now think that my comment might indeed be the complete answer in the case when $X$ has no zeroes.  

Guiseppe's answer has been a sort of Socratic catalyst.


Indeed, in that case the distribution $D$ defined by $\omega$ and $X^\flat$ agree. So $D$ is integrable if and only if the ideal generated by either $\omega$ or $X^\flat$ is differentiably closed, hence $dX^\flat = \alpha \wedge X^\flat$ for some one-form $\alpha$. In turn this is equivalent to $X^\flat \wedge dX^\flat = 0$, which is precisely the condition that $X$ be *twist-free*.


