Symmetries of the Grad-Shafranov Equation

A symmetry analysis of the Grad-Shafranov equation, for the standard case in which \textit{dP/d$\psi $} and \textit{dI/d$\psi $ }are constant, is presented. A Lie-group analysis reveals the full symmetry group, allowing for transformations of both independent and dependent variables. Several of the resulting symmetries appear to be new. They are used to construct a large family of exact group-invariant tokamak equilibria. This family includes the well known Solovev solution. The shape of the resulting flux surfaces and their stability are studied. The shape of the flux surfaces and stability of these solutions are analyzed.