Exactly solvable solution for the spherical to gamma-unstable transition in transitional nuclei is proposed by using the Bethe ansatz technique within an infinite-dimensional Lie algebra and dual algebraic structure. The duality relations between the unitary and quasi-spin algebraic structures for the boson and fermion systems are extended to the mixed boson-fermion system. The structure of U(6/4) nuclear supersymmetry scheme is discussed. We investigate the change in level structure induced by the phase transition by doing a quantal analysis. It is shown that the relation between the even-even and odd-A neighbors implied by nuclear supersymmetry in addition to dynamical symmetry limits can be also used for transitional regions. The experimental evidences are presented for even-even [E(5)] and odd-mass [E(5/4)] nuclei near the critical point symmetry. New experimental data on the 130Xe–131Xe and the 134Ba–135Ba super-multiplets were used to test the predictions of the supersymmetry scheme in the transition region. The low-states energy spectra for these nuclei have been also calculated and compared with the experimental data
