Conjecture on nonconstant proportionality factors admitting Beltrami fields

Let $U\subset\mathbb{R}^3$ be an open set, and let $f$ be a nonvanishing real analytic function on $U$ with $\nabla f\neq 0$ on $U$. A Beltrami field with proportionality factor $f$ is a vector field ${\mathbf u}$ solving the Beltrami system on $U$; we say that $f$ admits a nonzero Beltrami field if this system has a nonzero solution.

Beltrami-field proportionality conjecture. Nonconstant proportionality factors $f$ admitting a nonzero Beltrami field have the following properties:

• The space of proportionality factors $f$ admitting a nonzero Beltrami field is locally parametrized by 3 functions of 2 variables.
• If the level surfaces of $f$ contain no umbilic points, then $f$ admits at most a 2-dimensional space of Beltrami fields.
• A generic proportionality factor $f$ that admits a nonzero Beltrami field admits exactly a 1-dimensional space of Beltrami fields.

These claims refine the preceding dimension-counting results for Beltrami fields: the paper proves corresponding bounds in the totally umbilic and non-umbilic cases, while the asserted parametrization of admissible proportionality factors and the sharper generic dimension statements remain conjectural.