  The spherical to deformed $\gamma -unstable$ shape- phase transition in odd-A
nuclei is investigated by using the Dual algebraic structures and the affine
$SU(1,1)$ Lie Algebra within the framework of the interacting boson - fermion
model. The new algebraic solution for A-odd nuclei is introduced. In this
model, Single $j = 1/2 $ and $ 3/2 $ fermions are coupled with an even-even
boson core. Energy spectra, quadruple electromagnetic transitions and an
expectation value of the d-boson number operator are presented. Experimental
evidence for the $U^{BF} (5)-O^{BF} (6)$ transition in odd -A $Ba$ and $Rh$
isotopes is presented. The low-states energy spectra and $B(E2)$values for
these nuclei have been also calculated and compared with the experimental data.
