Hydromagnetic asymptotic equilibris with vacuum, closed-line dominant B and parallel dominant J

Asymptotic solutions of the magnetohydrostatic equations Δx(JxB)=0, ΔxB=0, J=ΔxB (with single-valued ∫dxxJxB) have been recently obtained to all orders in a torus with arbitrary section for a quasi-axisymmetric, quasi-azimuthal field configuration and quasi-vacuum B. In this paper it is considered a much weaker asymptotic ansatz, namely: i) the spatial domain where the solution is sought merely belongs to the torus homology class, ii) a vacuum, closed-line B dominates together with a parallel J; it is also proved that the corresponding problem of constructing an asymptotic solution can be reduced to solving a weakly singular Hammerstein integral equation to the lowest significant order and a weakly singular nonhomogeneous Fredholm integral equation of the 2nd kind of each of the subsequent orders