  A very approximate second integral of motion of the Dicke model is identified
within a broad region above the ground state, and for a wide range of values of
the external parameters. This second integral, obtained from a Born Oppenheimer
approximation, classifies the whole regular part of the spectrum in bands
labelled by its corresponding eigenvalues. Results obtained from this
approximation are compared with exact numerical diagonalization for finite
systems in the superradiant phase, obtaining a remarkable accord. The region of
validity of our approach in the parameter space, which includes the resonant
case, is unveiled. The energy range of validity goes from the ground state up
to a certain upper energy where chaos sets in, and extends far beyond the range
of applicability of a simple harmonic approximation around the minimal energy
configuration. The upper energy validity limit increases for larger values of
the coupling constant and the ratio between the level splitting and the
frequency of the field. These results show that the Dicke model behaves like a
two-degree of freedom integrable model for a wide range of energies and values
of the external parameters.
