  It is proved that the potentials of the form $\beta^{2n}$ (with $n$ being
integer) provide a ``bridge'' between the U(5) symmetry of the Bohr Hamiltonian
with a harmonic oscillator potential (occuring for $n=1$) and the E(5) model of
Iachello (Bohr Hamiltonian with an infinite well potential, materialized for
infinite $n$). Parameter-free (up to overall scale factors) predictions for
spectra and B(E2) transition rates are given for the potentials $\beta^4$,
$\beta^6$, $\beta^8$, corresponding to $R_4=E(4)/E(2)$ ratios of 2.093, 2.135,
2.157 respectively, compared to the $R_4$ ratios 2.000 of U(5) and 2.199 of
E(5). Hints about nuclei showing this behaviour, as well as about potentials
``bridging'' the E(5) symmetry with O(6) are briefly discussed. A note about
the appearance of Bessel functions in the framework of E(n) symmetries is given
as a by-product.
