On the orbits of magnetized Kepler problems in dimension 2k+12k+1

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension 2k+12k+1, the non-colliding orbits in the “external configuration space” R2k+1∖{0} are all conics; moreover, a conic orbit is an ellipse, a parabola, or a branch of a hyperbola according as the total energy is negative, zero, or positive. It is also demonstrated that the Lie group SO+(1,2k+1)×R+ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.