= Solution
The flux-function form solves $\nabla\mathbin\cdot\mathbf B=0$ identically. With $k=k(\psi)$, the poloidal relation $\rho\mathbf u_p=k\mathbf B_p$ and axisymmetry solve steady mass conservation. The ideal induction equation is solved by the velocity representation with the <field-line angular velocity> $\omega=\omega(\psi)$. The azimuthal momentum equation integrates once to the <magnetohydrodynamic angular-momentum invariant> $\ell=\ell(\psi)$.
What remains is the pressure or entropy equation and the two poloidal components of momentum. Projecting poloidal momentum along a field line produces the Bernoulli invariant in part (c); projecting across magnetic surfaces produces the transfield or Grad-Shafranov equation that determines their shape. An equation of state and boundary conditions complete the problem. The stellar gravitational potential is prescribed, so no self-gravity Poisson equation remains to solve.
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