Existence of global symmetries of divergence-free fields with first integrals

The relationship between symmetry fields and first integrals of divergence-free vector is explored in three dimensions light its relevance to plasma physics magnetic confinement fusion. A Noether-type theorem known: for each such symmetry, there corresponds a integral. extent which the converse true investigated. In doing so, reformulation this found holds on what called toroidal region. Some consequences methods presented are quick proofs existence flux coordinates high generality, without needing assume as cases magneto-hydrostatics or quasi-symmetry.