  Solvable supersymmetric algebraic model for descriptions of the spherical to
gama unstable shape- phase transition in even and odd mass nuclei is proposed.
This model is based on dual algebraic structure and Richardson - Gaudin method,
where the duality relations between the unitary and quasispin algebraic
structures for the boson and fermion systems are extended to mixed boson-
fermion system. The structure of two type of nuclear supersymmetry schemes,
based on the U(6/2) and U(6/4) supergroups, is discussed. We investigate the
change in level structure induced by the phase transition by doing a quantal
analysis. By using the generalized quasispin algebra, it is shown that the
nuclear supersymmetry concept can be also used for transitional regions in
addition to dynamical symmetry limits. Experimental evidence for the U(5)-O(6)
transition in Ru-Rh and Zn- Cu supermultiplets is presented. The low-states
energy spectra and B(E2)values for these nuclei have been calculated and
compared with the experimental data.
