 Parallel covector field

A one-form $k$ with ${\nabla }_b{k}_a=0$. If $\ell =d\varphi$ satisfies ${\nabla }_b{\ell }_a+{\ell }_a{\ell }_b=0$, then $k=e^{\varphi }\ell =d(e^{\varphi })$ is parallel.