Beltrami fields with nonconstant proportionality factor

We consider the question raised by Enciso and Peralta-Salas in [4] (see arXiv:1402.6825): What nonconstant functions $f$ can occur as the proportionality factor for a Beltrami field $\mathbf{u}$ on an open subset $U \subset \mathbb{R}^3$? We also consider the related question: For any such $f$, how large is the space of associated Beltrami fields? By applying Cartan's method of moving frames and the theory of exterior differential systems, we are able to improve upon the results given in [4]. In particular, the answer to the second question depends crucially upon the geometry of the level surfaces of $f$. We conclude by giving a complete classification of Beltrami fields that possess either a translation symmetry or a rotation symmetry.