Newton–Hooke-type symmetry of anisotropic oscillators

Rotation-less Newton–Hooke-type symmetry, found recently in the Hill problem, and instrumental for explaining the center-of-mass decomposition, is generalized to an arbitrary anisotropic oscillator in the plane. Conversely, the latter system is shown, by the orbit method, to be the most general one with such a symmetry. Full Newton–Hooke symmetry is recovered in the isotropic case. Star escape from a galaxy is studied as an application.

► Rotation-less Newton–Hooke (NH) symmetry is generalized to an arbitrary anisotropic oscillator.
► The orbit method is used to find the most general case for rotation-less NH symmetry.
► The NH symmetry is decomposed into Heisenberg algebras based on chiral decomposition.