  We show that a class of random all-to-all spin models, realizable in systems
of atoms coupled to an optical cavity, gives rise to a rich dynamical phase
diagram due to the pairwise separable nature of the couplings. By controlling
the experimental parameters, one can tune between integrable and chaotic
dynamics on the one hand, and between classical and quantum regimes on the
other hand. For two special values of a spin-anisotropy parameter, the model
exhibits rational-Gaudin type integrability and it is characterized by an
extensive set of spin-bilinear integrals of motion, independent of the spin
size. More generically, we find a novel integrable structure with conserved
charges that are not purely bilinear. Instead, they develop `dressing tails' of
higher-body terms, reminiscent of the dressed local integrals of motion found
in Many-Body Localized phases. Surprisingly, this new type of integrable
dynamics found in finite-size spin-1/2 systems disappears in the large-$S$
limit, giving way to classical chaos. We identify parameter regimes for
characterizing these different dynamical behaviors in realistic experiments, in
light of the limitations set by cavity dissipation.
