Abstract
The equations of ideal magnetohydrodynamics (MHD) with steady incompressible mass flow are known to be integrable to (1) a scalar elliptic partial differential equation (PDE) for a flux function in the case of helically symmetric systems; and (2) a system of two coupled scalar PDE’s in the case of arbitrary nontoroidal geometries. In the present paper it is shown that these integrated equations do not have the same symmetry properties as the original MHD equations; hence are not general enough. The appropriate generalizations are given, and their structure is analyzed. As a result a classification of stationary states of ideal MHD with flow in terms of isomorphisms to simpler MHD systems is obtained. Under certain mild conditions the involved differential operators can be mapped smoothly to considerably simpler ones; however, the mapping is not necessarily analytic. This transformation technique allows the construction of complicated MHD solutions with flow out of simpler MHD solutions. Some explicit examples are presented.
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