A generalization of Szebehely's equation for three dimensions

Szebehely's equation for the potential generating a prescribed family of orbits in two dimensions is generalized for three-dimensional orbits. A simultaneous system of first-order linear partial differential equations is derived for the determination of the potential in the three-dimensional case. Solutions of this system are found in several cases including Kepler's problem too.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.This paper is dedicated to Professor Victor Szebehely on the occasion of his 60th birthday.