Second prolongation of the rotation generator for plane graphs

The infinitesimal generator $V=u{\partial }_x-x{\partial }_u$ of rotations of the $(x,u)$ plane has second prolongationConsequently it preserves both $u_{xx}=0$ and $u_{xx}=(1+u_x^2{)}^{3/2}$, the equations for straight lines and consistently oriented unit-circle arcs.