Classical Bethe Ansatz and normal forms in an integrable version of the Dicke model

► We study the moment map of the Jaynes-Cummings model. ► We show that singularities are only of the elliptic and focus-focus types. ► We compute Normal forms using a classical version of Bethe Ansatz. ► We draw the bifurcation diagrams in the one spin and two spin models.