  We show that a Dirac Hamiltonian with equal scalar and vector harmonic
oscillator potentials has not only a spin symmetry but an U(3) symmetry and
that a Dirac Hamiltonian with scalar and vector harmonic oscillator potentials
equal in magnitude but opposite in sign has not only a pseudospin symmetry but
a pseudo-U(3) symmetry. We derive the generators of the symmetry for each case.
