  We derive the Holstein-Primakoff oscillator realization on the coadjoint
orbits of the $SU(N+1)$ and $SU(1,N)$ group by treating the coadjoint orbits as
a constrained system and performing the symplectic reduction. By using the
action-angle variables transformations, we transform the original variables
into Darboux variables. The Holstein-Primakoff expressions emerge after
quantization in a canonical manner with a suitable normal ordering. The
corresponding Dyson realizations are also obtained and some related issues are
discussed.
