The utility of differential forms for understanding the origin of the self-dual gauge-field equations is illustrated by deriving the systems of linear partial differential equations introduced by Belavin and Zakharov and used in a different form by Ueno and Nakamura. The integrability condition for these systems of equations is then used to show their relation to a generalized form of the Ernst equation.
Physics Essays 3, 406-413 (1990)
J. Appl. Phys. Vol. 68, p. 3818 (1990) (PDF)
The general approach to cylindrically symmetric force-free magnetic fields first introduced by Lust and Schluter [Z. Astrophys. Vol. 34, 263 (1954)], is restricted to cylindrically symmetric fields, and subsequently used to determine a set of solutions to the force-free field equations with non-constant alpha. The first element of the set is the well known constant a solution of Lundquist [Ark. Fys. Vol. 2, 361 (1951)]. These solutions may have practical applications with respect to high-temperature superconductors.