We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of N sites connected at its ends to two reservoirs can be solved exactly, i.e., the factorial moments of the non-equilibrium steady-state can be written in the closed form for each N. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: (i) the introduction of a dual absorbing process reducing the problem to a finite number of particles and (ii) the solution of the dual dynamics exploiting a symmetry obtained from the quantum inverse scattering method. Long-range correlations are computed in the finite-volume system. The exact solution allows us to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.
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