  Analytical solutions of the Bohr Hamiltonian are obtained in the
$\gamma$-unstable case, as well as in an exactly separable rotational case with
$\gamma\approx 0$, called the exactly separable Morse (ES-M) solution. Closed
expressions for the energy eigenvalues are obtained through the Asymptotic
Iteration Method (AIM), the effectiveness of which is demonstrated by solving
the relevant Bohr equations for the Davidson and Kratzer potentials. All medium
mass and heavy nuclei with known $\beta_1$ and $\gamma_1$ bandheads have been
fitted by using the two-parameter $\gamma$-unstable solution for transitional
nuclei and the three-parameter ES-M for rotational ones. It is shown that
bandheads and energy spacings within the bands are well reproduced for more
than 50 nuclei in each case.
