  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 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; 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 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.
