  We derive the exact solution of a system of N-level atoms in contact with a
Markovian reservoir. The resulting Liouvillian expressed in a vectorized basis
is mapped to an SU(N) trigonometric Richardson-Gaudin model whose exact
solution for the complete set of eigenmodes is given by a set of non-linear
coupled equations. For N = 2 (SU(2)) we recover the exact solution of Phys.
Rev. Lett. 122, 010401 (2019). We then study the SU(3) case for three-level
atom systems and discuss the properties of the steady state and dissipative
gaps for finite systems as well as for the thermodynamic limit.
