Let $X$ and $Y$ be complete commuting vector fields on a (semi-) Riemannian manifold.
Denote by $\pi X$ the component of $X$ orthogonal to $Y$, i.e.
$\pi X = X - \frac{\langle X,Y\rangle}{\langle Y,Y\rangle} Y$.
Can we say anything about whether $\pi X$ is complete?