  In specific open systems with collective dissipation the Liouvillian can be
mapped to a non-Hermitian Hamiltonian. We here consider such a system where the
Liouvillian is mapped to an XXZ Richardson-Gaudin integrable model and detail
its exact Bethe ansatz solution. While no longer Hermitian, the Hamiltonian is
pseudo-Hermitian/PT-symmetric, and as the strength of the coupling to the
environment is increased the spectrum in a fixed symmetry sector changes from a
broken pseudo-Hermitian phase with complex conjugate eigenvalues to a
pseudo-Hermitian phase with real eigenvalues, passing through a series of
exceptional points and associated dissipative quantum phase transitions. The
homogeneous limit supports a nontrivial steady state, and away from this limit
this state gives rise to a slow logarithmic growth of the decay rate (spectral
gap) with system size. Using the exact solution, it is furthermore shown how at
large coupling strengths the ratio of the imaginary to the real part of the
eigenvalues becomes approximately quantized in the remaining symmetry sectors.
