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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 toa semi-classical approach of the model Hamiltonian while the other one providesthe exact eigenvalues. Both procedures yield close formulas for energies. Thefirst procedure involves four parameters, while the second involves a compactformula with five parameters. In each case the parameters are fixed by aleast-square fit procedure. Applications are performed for eight even-evennuclei.  Both methods yield results which are in a surprisingly good agreement withthe experimental data. We give also our predicted reduced transitionprobabilities within the two approaches, although the correspondingexperimental data are not yet available.

Phenomenological description of the states $0^+$ and $2^+$ in some even-even nuclei