Eigenstates and eigenenergies of four-wave-mixing models

We present explicit analytical expressions of all the eigenenergies and eigenstates for two different kinds of four-wave-mixing model in terms of a parameter lambda without the so-called Bethe ansatz assumption. The parameter lambda is shown to be determined by the roots of a simple polynomial. We also obtain the exact explicit analytical expressions of infinite eigenstates and energies without any unknown parameter.