  The symmetry properties of a classical N-dimensional harmonic oscillator with
rational frequency ratios are studied from a global point of view. A
commensurate oscillator possesses the same number of globally defined constants
of motion as an isotropic oscillator. In both cases invariant phase-space
functions form the algebra su(N) with respect to the Poisson bracket. In the
isotropic case, the phase-space flows generated by the invariants can be
integrated globally to a set of finite transformations isomorphic to the group
SU(N). For a commensurate oscillator, however, the group SU(N) of symmetry
transformations is found to exist only on a reduced phase space, due to
unavoidable singularities of the flow in the full phase space. It is therefore
crucial to distinguish carefully between local and global definitions of
symmetry transformations in phase space. This result solves the longstanding
problem of which symmetry to associate with a commensurate harmonic oscillator.
