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We investigate the planar anisotropic harmonic oscillator with explicitrotational symmetry as a particle model with non-commutative coordinates. Itincludes the exotic Newton-Hooke particle and the non-commutative Landauproblem as special, isotropic and maximally anisotropic, cases. The system isdescribed by the same (2+1)-dimensional exotic Newton-Hooke symmetry as in theisotropic case, and develops three different phases depending on the values ofthe two central charges. The special cases of the exotic Newton-Hooke particleand non-commutative Landau problem are shown to be characterized by additional,so(3) or so(2,1) Lie symmetry, which reflects their peculiar spectralproperties.

Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton-Hooke symmetry