The Levi-Civita Time Tranformation and a Killing Vector for the Kepler Problem

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