A characterization of the Ligon-Schaaf regularization map

First, we give an explicit description of all the mappings from the phace space of the Kepler problem to the phase space of the geodesics on the sphere, which transform the constants of motion of the Kepler problem to the angular momentum. Second, among these we describe those mappings which in addition send Kepler solutions to parametrized geodesics. Third, we describe those mappings which in addition are canonical transformations of the respective phase space. Finally we prove that among these the Ligon-Schaaf map is the unique one which maps the collison orbits to the geodesics which pass through the north pole. In this way we also give a new proof that the Ligon-Schaaf map has all the properties described above