  We study the ergodic -- non-ergodic transition in a generalized Dicke model
with independent co- and counter rotating light-matter coupling terms. By
studying level statistics, the average ratio of consecutive level spacings, and
the quantum butterfly effect (out-of-time correlation) as a dynamical probe, we
show that the ergodic -- non-ergodic transition in the Dicke model is a
consequence of the proximity to the integrable limit of the model when one of
the couplings is set to zero. This can be interpreted as a hint for the
existence of a quantum analogue of the classical Kolmogorov-Arnold-Moser
theorem. Besides, we show that there is no intrinsic relation between the
ergodic -- non-ergodic transition and the precursors of the normal --
superradiant quantum phase transition.
