A Note on the First Integrals of Vector Fields with Integrating Factors and Normalizers

We prove a sufficient condition for the existence of explicit first integrals for vector fields which admit an integrating factor. This theorem recovers and extends previous results in the literature on the integrability of vector fields which are volume preserving and possess nontrivial normalizers. Our approach is geometric and coordinate-free and hence it works on any smooth orientable manifold

Beltrami fields with a nonconstant proportionality factor are rare

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.Comment: 13 page