  Schroedinger bound-state problem in D dimensions is considered for a set of
central polynomial potentials (containing 2q coupling constants). Its
polynomial (harmonic-oscillator-like, quasi-exact, terminating) bound-state
solutions of degree N are sought at a (q+1)-plet of exceptional
couplings/energies, the values of which comply with (the same number of)
termination conditions. We revealed certain hidden regularity in these coupled
polynomial equations and in their roots. A particularly impressive
simplification of the pattern occurred at the very large spatial dimensions D
where all the "multi-spectra" of exceptional couplings/energies proved
equidistant. In this way, one generalizes one of the key features of the
elementary harmonic oscillators to (presumably, all) non-vanishing integers q.
