From extensive numeric diagonalizations of the SU(3) Perk-Schultz Hamiltonian with a special value of the anisotropy and different boundary conditions, we have observed simple regularities for a significant part of its eigenspectrum. In particular the ground state energy and nearby excitations belong to this part of the spectrum. Our simple formulae describing these regularities remind, apart from some selection rules, the eigenspectrum of the free fermion model. Based on the numerical

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... ons we formulate several conjectures. Using explicit solutions of the associated nested Bethe-ansatz equations, guessed from an analysis of the functional equations of the model, we provide evidence for a part of them.