The Levi-Civita time transformation and a killing vector for the Kepler problem

As an outcome of our hypercomplex unitary description [9] of some peculiar aspects of the Kepler problem (regularization, prequantization of the Kepler manifold), we show how both the differential time transformation adopted by Levi-Civita for the regularization of the plane problem, and the quaternionic transformation which prequantizes the Kepler manifold, derive from the same quaternionic definition of a Keplerian orbit. Moreover, we show how the stereographic projection of an hodographical sphere leads to a Killing vector, the counterpart of the position vector of the particle. Interesting results are brought out.