  A sixth-order quadrupole boson Hamiltonian is used to describe the states
$0^+$ and $2^+$ identified in several nuclei by various types of experiments.
Two alternative descriptions of energy levels are proposed. One corresponds to
a semi-classical approach of the model Hamiltonian while the other one provides
the exact eigenvalues. Both procedures yield close formulas for energies. The
first procedure involves four parameters, while the second involves a compact
formula with five parameters. In each case the parameters are fixed by a
least-square fit procedure. Applications are performed for eight even-even
nuclei.
  Both methods yield results which are in a surprisingly good agreement with
the experimental data. We give also our predicted reduced transition
probabilities within the two approaches, although the corresponding
experimental data are not yet available.
