Symmetry transformations for magnetohydrodynamics and Chew–Goldberger–Low equilibria revisited

We generalize the symmetry transformations for magnetohydrodynamic (MHD) equilibria with isotropic pressure and incompressible flow parallel to the magnetic field introduced by Bogoyavlenskij in the case of the respective Chew–Goldberger–Low (CGL) equilibria with anisotropic pressure. We find that the geometrical symmetry of the field-aligned equilibria can be broken by those transformations only when the magnetic field is purely poloidal. In this situation we derive three-dimensional CGL equilibria from given axisymmetric ones. Also, we examine the generic symmetry transformations for MHD and CGL equilibria with incompressible flow of an arbitrary direction, introduced in a number of papers, and find that they cannot break the geometrical symmetries of the original equilibria, unless the velocity and magnetic field are collinear and purely poloidal.