  The 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 application.
