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The wave map scaling symmetry is ${\varphi }_{\lambda }(t,x)=\varphi (t/\lambda ,x/\lambda )$. Its conserved wave map energy isThus energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\ge 3$. The scaling-critical homogeneous Sobolev spaces for perturbations of a constant map are ${\dot{H}}^{n/2}\times {\dot{H}}^{n/2-1}$. These relations constitute wave map energy and criticality.

Put $q=u/x^2$, so $F=xG(q)$ andSubstitution into the Lie-symmetry determining equation for a first-order ordinary differential equation and separation with respect to the arbitrary function $G$ givesThese equations imply $\xi =Cx$ and $\eta =2Cu$. Apart from the zero generator, the common symmetry for arbitrary $G$ is therefore generated byIts one-parameter group is the scaling symmetrywhich leaves $u/x^2$ invariant. Special choices of $G$ can have additional symmetries, but this is the complete symmetry valid for an arbitrary smooth $G$.