  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.
