Let $M$ be a 4-dimensional Riemannian manifold. Let $\kappa$ be a 1-form.
I look for solution function $\phi$, such that there exists functions $\alpha$ and $\beta$ \begin{equation} {\left( {d\phi \wedge \kappa } \right)^ + } = \alpha d{\kappa ^ + },\;\;\;\;{\left( {d\phi \wedge \kappa } \right)^ - } = \beta {\left( {d\kappa } \right)^ - } \end{equation}
Obviously, constant function $\phi$ is one solution, where $\alpha$ and $\beta$ are both zero. But I wonder if there are more solutions in general, and how to construct them?
Example
On $S^4$, picking $\kappa = {\sin ^2}\rho \left( {{{\cos }^2}\theta d{\varphi _1} + {{\sin }^2}\theta d{\varphi _2}} \right)$, with spherical coordinate \begin{equation} \begin{gathered} {x_0} = \cos \rho \hfill \\ {x_1} = \sin \rho \cos \theta \cos {\varphi _1} \hfill \\ {x_2} = \sin \rho \cos \theta \sin {\varphi _1} \hfill \\ {x_3} = \sin \rho \sin \theta \cos {\varphi _2} \hfill \\ {x_4} = \sin \rho \sin \theta \sin {\varphi _2} \hfill \\ \end{gathered} \end{equation}
one can find a non-trivial solution $\phi \sim \cos \rho$.