Second constant of motion for two-dimensional positronium in a magnetic field

Recent numerical work indicates that the classical motion of positronium in a constant magnetic field does not exhibit chaotic behavior if the system is confined to two dimensions. One would therefore expect this system to possess a second constant of the motion in addition to the total energy. In this paper we construct a generalization of the Laplace-Runge-Lenz vector and show that a component of this vector is a constant of the motion.