Generalized Kustaanheimo–Stiefel transformations

In this paper the theory of constructing the generalized KS transformations is given for the Kepler problem dimensions $q+1$ ($q=2^h$, $h=0,1,2,\dots$). The following theorem is proved: The connection between the Kepler problem in $(q+1)$-dimensional real space and the problem of an isotropic harmonic oscillator in real space of dimension $N$ exists and can be established by using the generalized KS transformations only for the cases, when $N=2q$ and $q=2^h$ ($h=0,1,2,\dots$). A simple graphic method of constructing the generalized KS transformations realizing this connection is also suggested.