  Starting from the original collective Hamiltonian of Bohr and separating the
beta and gamma variables as in the X(5) model of Iachello, an exactly soluble
model corresponding to a harmonic oscillator potential in the beta-variable (to
be called X(5)-$\beta^2$) is constructed. Furthermore, it is proved that the
potentials of the form $\beta^{2n}$ (with n being integer) provide a ``bridge''
between this new X(5)-$\beta^2$ model (occuring for n=1) and the X(5) model
(corresponding to an infinite well potential in the beta-variable, materialized
for n going to infinity. Parameter-free (up to overall scale factors)
predictions for spectra and B(E2) transition rates are given for the potentials
$\beta^2$, $\beta^4$, $\beta^6$, $\beta^8$, corresponding to E(4)/E(2) ratios
of 2.646, 2.769, 2.824, and 2.852 respectively, compared to the E(4)/E(2)
ratios of 2.000 for U(5) and 2.904 for X(5). Hints about nuclei showing this
behaviour, as well as about potentials ``bridging'' the X(5) symmetry with
SU(3) are briefly discussed.
