Magnetic field integrability and the current closure condition

Magnetic fields with translational symmetry, axisymmetry, and helical symmetry are integrable. That is, they have good flux surfaces. The current closure condition, that contour-integral dl/B is a flux surface quantity, also holds. It is possible to construct magnetic fields with broken symmetry, which have nearly integrable magnetic fields. It is shown that for nearly integrable nonsymmetric fields, the current closure condition need not hold