  We demonstrate how the presence of continuous weak symmetry can be used to
analytically diagonalize the Liouvillian of a class Markovian dissipative
systems with arbitrary strong interactions or nonlinearity. This enables an
exact description of the full dynamics and dissipative spectrum. Our method can
be viewed as implementing an exact, sector-dependent mean-field decoupling, or
alternatively, as a kind of quantum-to-classical mapping. We focus on two
canonical examples: a nonlinear bosonic mode subject to incoherent loss and
pumping, and an inhomogeneous quantum Ising model with arbitrary connectivity
and local dissipation. In both cases, we calculate and analyze the full
dissipation spectrum. Our method is applicable to a variety of other systems,
and could provide a powerful new tool for the study of complex
driven-dissipative quantum systems.
