I have this sort of research project about symmetries under the central potential and I'm stuck on this Runge-Lenz vector. As it is a conserved quantity I was expecting it to come out of Noether's theorem. I can't figure out how. So I go on the net to find out and get 2 answers: infinitesimal Lorentz trasnformation without rotation followed by a time translation and a more explicit article on a canonical transformation. Out of the latter I find that besides the usual space-time transformations come the so called kinematical symmetries, while out of symmetries in the phase space comes another kind of conserved quantities called dynamical. It shows rather clearly that the angle the LRL vector makes with the x axis is the canonical conjugate of L and that's the pair of variables I should canonical transform to. What other dynamical symmetries are there in the world that I heven't heared of and how should the conserved quantities look in general terms? Is there an analogue to Noether's theorem that would tell me exactly how the conserved quantity should look like? How did Laplace, Runge and Lenz get the expression for the vector?