 Cartesian magnetostatic flux-function equilibrium

In a $y$-independent magnetostatic equilibrium with no axial pressure or gravitational force, $\nabla \psi \times \nabla B_y=0$. Thus $B_y=b(\psi )$ on a regular connected flux region. The remaining balance is $[{\nabla }^2\psi +b{b}^{\prime }]\nabla \psi /{\mu }_0+\nabla p+\rho \nabla \Phi =0$. Constancy on separate components of a level set does not automatically give one global single-valued $b$.