  We analytically investigate a 1d branching-coalescing model with reflecting
boundaries in a canonical ensemble where the total number of particles on the
chain is conserved. Exact analytical calculations show that the model has two
different phases which are separated by a second-order phase transition. The
thermodynamic behavior of the canonical partition function of the model has
been calculated exactly in each phase. Density profiles of particles have also
been obtained explicitly. It is shown that the exponential part of the density
profiles decay on three different length scales which depend on total density
of particles.
