MICZ-Kepler = dynamics on the cone over the rotation group

We show that the n-dimensional MICZ-Kepler system arises from symplectic reduction of a simple mechanical system on the cone over the rotation group SO(n). As a corollary we derive an elementary formula for its general solution. The punch-line of our computation is that the additional MICZ-Kepler $|\phi|^2/r^2$ type potential term is the rotational part of the cone's kinetic energy.