The harmony in the Kepler and related problems

The technique of reduction of order developed by Nucci [J. Math. Phys. 37, 1772-1775 (1996)] is used to produce nonlocal symmetries in addition to those reported by Krause [J. Math. Phys. 35, 5734-5748 (1994)] in his study of the complete symmetry group of the Kepler problem. The technique is shown to be applicable to related problems containing a drag term which have been used to model the motion of low altitude satellites in the Earth's atmosphere and further generalizations. A consequence of the application of this technique is the demonstration of the group theoretical relationship between the simple harmonic oscillator and the Kepler and related problems.