Reduction of the classical MICZ-Kepler problem to a two-dimensional linear isotropic harmonic oscillator

The classical MICZ-Kepler problem is shown to be reducible to an isotropic two-dimensional system of linear harmonic oscillators and a conservation law in terms of new variables related to the Ermanno-Bernoulli constants and the components of the Poincare vector. An algorithmic route to linearization is shown based on Lie symmetry analysis and the reduction method [Nucci, J. Math. Phys. 37, 1772 (1996) ]. First integrals are also obtained by symmetry analysis and the reduction method [Marcelli and Nucci,J. Math. Phys. 44, 2111 (2002) ].