  We study nuclear matter and finite nuclei with a chiral Lagrangian which
generalizes the linear $\sigma$ model and also accounts for the QCD trace
anomaly by means of terms which involve the $\sigma$ and $\vmg{\pi}$ fields as
well as the glueball field $\phi$. The form of the scale invariant term leading
to an omega meson mass, after symmetry breaking, could involve coupling to
$\phi^2$ or $\sigma^2$ or some linear combination thereof. In fact an
$\omega_{\mu}\omega^{\mu}\phi^2$ form is strongly favored by the bulk
properties of nuclei, which also rather strongly constrain the other
parameters. A reasonable description of the closed shell nuclei oxygen, calcium
and lead can be achieved, although the spin-orbit splittings are somewhat
smaller than the experimental values.
