  Exactly solvable solution for the spherical to gamma - unstable transition in
transitional nuclei based on dual algebraic structure and nuclear supersymmetry
concept is proposed. The duality relations between the unitary and quasispin
algebraic structures for the boson and fermion systems are extended to mixed
boson-fermion system. 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.
