  We consider the existence of Beltrami fields with a nonconstant
proportionality factor $f$ in an open subset $U$ of $\mathbf{R}^3$. By
reformulating this problem as a constrained evolution equation on a surface, we
find an explicit differential equation that $f$ must satisfy whenever there is
a nontrivial Beltrami field with this factor. This ensures that there are no
nontrivial solutions for an open and dense set of factors $f$ in the $C^k$
topology. In particular, there are no nontrivial Beltrami fields whenever $f$
has a regular level set diffeomorphic to the sphere. This provides an
explanation of the helical flow paradox of Morgulis, Yudovich and Zaslavsky.
