Suppose $F:U \to \mathbb{R}^n$ is a $\mathcal{C}^1$ vector field on an open set $U \subseteq \mathbb{R}^n$. Let $\lambda \in \mathbb{C}$. Consider the partial differential equation
$$F(x) \cdot \nabla \psi (x) = \lambda \psi (x).$$
Under what conditions is a solution $\psi: U \to \mathbb{C}$ guaranteed to exist? If a solution exists, under what conditions is it unique?
Does the answer admit further refinement under the additional assumptions that $F$ is a complete vector field and that there is a smooth attractor in $U$ such that all integral curves of $F$ in $U$ asymptotically approach this attractor in positive time?