The classical mechanics of two Coulomb charges on a plane (e1, m1) and (e2, m2) subject to a constant magnetic field perpendicular to the plane is considered. Special "superintegrable" trajectories (circular and linear) for which the distance between charges remains unchanged are indicated and their constants of motion are presented. The number of the independent constants of motion for the special trajectory is larger than for generic ones and hence they can be called "superintegrable." A classification of pairs of charges for which special trajectories occur is given. The special trajectories are analyzed for three particular cases, namely that of two electrons, an electron-positron pair, and an electron-α-particle pair.
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