A similarity reduction of the Grad-Shafranov equation

A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad-Shafranov equation. As an illustration of the method, the frequently used Solov'ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the Grad-Shafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions.