  The spectral properties of a tractable collective model Hamiltonian are
studied. The potential energy is truncated up to quartic terms in the
quadrupole deformation variables, incorporating vibrational,
$\gamma$-independent rotational and axially deformed rotational structures.
These physically significant limits are analysed in detail and confronted with
well-established approximation schemes. Furthermore, transitional Hamiltonians
in between the limits are presented and discussed. All results are obtained
within a recently presented Cartan-Weyl based framework to calculate
$SU(1,1)\times SO(5)$ embedded quadrupole collective observables.
