  We study a one-parameter generalization of the symmetric simple exclusion
process on a one dimensional lattice. In addition to the usual dynamics (where
particles can hop with equal rates to the left or to the right with an
exclusion constraint), annihilation and creation of pairs can occur. The system
is driven out of equilibrium by two reservoirs at the boundaries. In this
setting the model is still integrable: it is related to the open XXZ spin chain
through a gauge transformation. This allows us to compute the full spectrum of
the Markov matrix using Bethe equations. Then, we derive the spectral gap in
the thermodynamical limit. We also show that the stationary state can be
expressed in a matrix product form permitting to compute the multi-points
correlation functions as well as the mean value of the lattice current and of
the creation-annihilation current. Finally the variance of the lattice current
is exactly computed for a finite size system. In the thermodynamical limit, it
matches perfectly the value obtained from the associated macroscopic
fluctuation theory. It provides a confirmation of the macroscopic fluctuation
theory for dissipative system from a microscopic point of view.
