Lie-symmetry methods are used to determine the symmetry group of reduced magnetohydrodynamics. This group allows for arbitrary, continuous transformations of the fields themselves, along with space-time transformations. The derivation reveals, in addition to the predictable translation and rotation groups, some unexpected symmetries. It also uncovers novel, exact nonlinear solutions to the reduced system. A similar analysis of a related but simpler system, describing nonlinear plasma turbulence in terms of a single field, is also presented.
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