  We develop a uniform semiclassical trace formula for the density of states of
a three-dimensional isotropic harmonic oscillator (HO), perturbed by a term
$\propto r^4$. This term breaks the U(3) symmetry of the HO, resulting in a
spherical system with SO(3) symmetry. We first treat the anharmonic term in
semiclassical perturbation theory by integration of the action of the perturbed
periodic HO orbits over the manifold $\mathbb{C}$P$^2$ which characterizes
their 4-fold degeneracy. Then we obtain an analytical uniform trace formula
which in the limit of strong perturbations (or high energy) asymptotically goes
over into the correct trace formula of the full anharmonic system with SO(3)
symmetry, and in the limit $\epsilon$ (or energy) $\to 0$ restores the HO trace
formula with U(3) symmetry. We demonstrate that the gross-shell structure of
this anharmonically perturbed system is dominated by the two-fold degenerate
diameter and circular orbits, and {\it not} by the orbits with the largest
classical degeneracy, which are the three-fold degenerate tori with rational
ratios $\omega_r:\omega_\phi=N:M$ of radial and angular frequencies. The same
holds also for the limit of a purely quartic spherical potential $V(r)\propto
r^4$.
