  Davidson potentials of the form $\beta^2 +\beta_0^4/\beta^2$, when used in
the original Bohr Hamiltonian for $\gamma$-independent potentials bridge the
U(5) and O(6) symmetries. Using a variational procedure, we determine for each
value of angular momentum $L$ the value of $\beta_0$ at which the derivative of
the energy ratio $R_L=E(L)/E(2)$ with respect to $\beta_0$ has a sharp maximum,
the collection of $R_L$ values at these points forming a band which practically
coincides with the ground state band of the E(5) model, corresponding to the
critical point in the shape phase transition from U(5) to O(6). The same
potentials, when used in the Bohr Hamiltonian after separating variables as in
the X(5) model, bridge the U(5) and SU(3) symmetries, the same variational
procedure leading to a band which practically coincides with the ground state
band of the X(5) model, corresponding to the critical point of the U(5) to
SU(3) shape phase transition. A new derivation of the Holmberg-Lipas formula
for nuclear energy spectra is obtained as a by-product.
