The relationship between symmetry fields and first integrals of
divergence-free vector fields is explored in three dimensions in light of its
relevance to plasma physics and magnetic confinement fusion. A Noether-type
Theorem is known: for each such symmetry, there corresponds a first integral.
The extent to which the converse is true is investigated. In doing so, a
reformulation of this Noether-type Theorem is found for which the converse
holds on what is called the toroidal region. Some consequences of the methods
presented are quick proofs of the existence of flux coordinates for magnetic
fields in high generality; without needing to assume a symmetry such as in the
cases of magneto-hydrostatics (MHS) or quasi-symmetry.Comment: 31 pages, 3 figures. This version of the article features an example
involving Reeb cylinders whose idea was suggested by Daniel Peralta-Sala