  Hill's equations, which first arose in the study of the Earth-Moon-Sun
system, admit the two-parameter centrally extended Newton-Hooke symmetry
without rotations. This symmetry allows for extending Kohn's theorem about the
center-of-mass decomposition. Particular light is shed on the problem using
Duval's "Bargmann" framework. The separation of the center-of-mass motion into
that of a guiding center and relative motion is derived by a generalized chiral
decomposition.
