Symmetry of magnetic field and the existence of global Clebsch coordinates

Darboux's theorem guarantees that a magnetic field B(x) in R³ can always be written as B(x) =∇α×∇β for two locally-defined scalar functions α(x) and β(x), which are known as the Clebsch coordinates. The local Clebsch coordinates exist even when the magnetic field is chaotic. However, global Clebsch coordinates do not exist in general. We prove that a necessary and sufficient condition for the existence of global Clebsch coordinates over a contractible domain is that the magnetic field admits a global symmetry, i.e., there exists a vector field η such that the Lie derivative of the 1-form potential A(x) along η is exact. When the symmetry exists, let L_η dA=dS and we can construct the global Clebsch coordinates for B(x) over contractible subsets of the level sets of the function A·η-S. It turns out that this result is implied by the Poincaré lemma on 2D contractible surfaces.