  The Hilbert-Schmidt operator formulation of non-commutative quantum mechanics
in 2D Moyal plane is shown to allow one to construct Schwinger's SU(2)
generators. Using this the SU(2) symmetry aspect of both commutative and
non-commutative harmonic oscillator are studied and compared. Particularly, in
the non-commutative case we demonstrate the existence of a critical point in
the parameter space of mass and angular frequency where there is a manifest
SU(2) symmetry for a unphysical harmonic oscillator Hamiltonian built out of
commuting (unphysical yet covariantly transforming under SU(2)) position like
observable. The existence of this critical point is shown to be a novel aspect
in non-commutative harmonic oscillator, which is exploited to obtain the
spectrum and the observable mass and angular frequency parameters of the
physical oscillator-which is generically different from the bare parameters
occurring in the Hamiltonian. Finally, we show that a Zeeman term in the
Hamiltonian of non-commutative physical harmonic oscillator, is solely
responsible for both SU(2) and time reversal symmetry breaking.
