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.

twist-free; although this condition seems stronger. Recall that a Killing vector $X$ is twist-free if $X^\flat \wedge dX^\flat = 0$, where $X^\flat = g(X,\cdot)$ in your notation. – José Figueroa-O'Farrill Mar 2 '11 at 13:00