 Characteristic components of a hyperbolic first-order equation

The characteristic ordinary differential equations $\dot{x}=y$, $\dot{y}=x$ preserve $u=x^2-y^2$. Away from the origin, $u=x^2-y^2$, $v=2xy$ have nonzero Jacobian determinant $4(x^2+y^2)$, and the chain rule reduces the displayed equation to $2(x^2+y^2)f_v=0$. Hence locally $f=F(u)$.

Globally, a level set of $u$ can have disconnected characteristic components, permitting different profiles on its different branches. A smooth counterexample on the whole plane is $f=\exp [-1/(x^2-y^2{)}^2]$ where $x>0$ and $x^2>y^2$, and $f=0$ elsewhere. The exponential is flat on the wedge boundary, so this extension is smooth and satisfies the equation. Nevertheless the points $(1,0)$ and $(-1,0)$ have equal $u$ and different function values. One global profile $F(u)$ therefore needs an additional branch-identification condition, rather than following from the local change of variables alone.