  We discuss a sufficient condition for gapless excitations in the Lindbladian
master equation for collective spin-boson systems and permutationally invariant
systems. The condition relates a nonzero macroscopic cumulant correlation in
the steady state to the presence of gapless modes in the Lindbladian. In phases
arising from competing coherent and dissipative Lindbladian terms, we argue
that such gapless modes, concomitant with angular momentum conservation, can
lead to persistent dynamics in the spin observables with the possible formation
of dissipative time-crystals. We study different models within this
perspective, from Lindbladians with Hermitian jump operators, to non-Hermitian
ones composed by collective spins and Floquet spin-boson systems. We also
provide a simple analytical proof for the exactness of mean-field semiclassical
approach in such systems based on a cumulant expansion.
