  The structure of static MHD equilibria that admit continuous families of
Euclidean symmetries is well understood. Such field configurations are governed
by the classical Grad-Shafranov equation, which is a single elliptic PDE in two
space dimensions. By revealing a hidden symmetry, we show that in fact all
smooth solutions of the equilibrium equations with non-vanishing pressure
gradients away from the magnetic axis satisfy a generalization of the
Grad-Shafranov equation. In contrast to solutions of the classical
Grad-Shafranov equation, solutions of he generalized equation are not
automatically equilibria, but instead only satisfy force balance averaged over
the one-parameter hidden symmetry. We then explain how the generalized
Grad-Shafranov equation can be used to reformulate the problem of finding exact
three-dimensional smooth solutions of the equilibrium equations as finding an
optimal volume-preserving symmetry.
