Generalized KS transformations, $ND$ singular oscillator and generalized MICZ-Kepler system
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that N-dimensionall singular oscillator and (n+1)-dimensional generalized MICZ-Kepler system are dual to each other and the duality transformation is the generalized version of the KS transformation. The solvability of the Schrodinger equation of these problems by the variables separation method is given in a double, spherical and parabolic coordinates. The quadratic Hahn algebra QH(3) as a hidden symmetry remains unchanged with different decomposition of the original real space ${\rm I \!R}^{N}$ into components in the framework of addition rule for SU(1,1) algebra. Also the hidden symmetry algebra as Higgs/Hahn algebras is clearly shown by the commutant approach in the sense of Howe duality. A dimensional reduction is carried out to a singular oscillator of two dimensions where every variable is the n-dimensional hyper-radius r. The dual connection with (N=2n)-dimensional singular oscillator and the (n+1) general MICZ-Kepler system in the class of quasi-exact (QE) problems also are considered. The certain generalization both of harmonic oscillator model by anisotropic and nonlinear inharmonic terms and its dual analog is shown and analyzed in the framework of generalized KS transformations. The exact analytical solutions of the Schrodinger equation for abovementioned problems for QE class are discused and given for four series of dual quasi-exact solvable models. In particular, a comparison with similar results in lower dimensions and its generalization are given.