  The aim of this two-part paper is to investigate the stability properties of
a special class of solutions to a coagulation-fragmentation equation. We assume
that the coagulation kernel is close to the diagonal kernel, and that the
fragmentation kernel is diagonal. We construct a two-parameter family of
stationary solutions concentrated in Dirac masses. We carefully study the
asymptotic decay of the tails of these solutions, showing that this behaviour
is stable. In a companion paper we prove that for initial data which are
sufficiently concentrated, the corresponding solutions approach one of these
stationary solutions for large times.
