  Motivated by recent prototypes of engineered atomic spin devices, we study a
fully connected system of $N$ spins $1/2$, modeled by the Lipkin-Meshkov-Glick
(LMG) model of a collective spin $s=N/2$ in the presence of Markovian
dissipation processes. We determine and classify the different phases of the
dissipative LMG model with Markovian dissipation, including the properties of
the steady-state and the dynamic behavior in the asymptotic long-time regime.
Employing variational methods and a systematic approach based on the
Holstein-Primakoff mapping, we determine the phase diagram and the spectral and
steady-state properties of the Liouvillian by studying both the infinite-$s$
limit and $1/s$ corrections. Our approach reveals the existence of different
kinds of dynamical phases and phase transitions, multi-stability and regions
where the dynamics is recurrent. We provide a classification of critical and
non-critical Liouvillians according to their spectral and steady-state
properties.
