  Hirota's bilinear method ("direct method") has been very effective in
constructing soliton solutions to many integrable equations. The construction
of one- and two-soliton solutions is possible even for non-integrable bilinear
equations, but the existence of a generic three-soliton solution imposes severe
constraints and is in fact equivalent to integrability. This property has been
used before in searching for integrable partial differential equations, and in
this paper we apply it to two dimensional partial difference equations defined
on a 3x3 stencil. We also discuss how the obtained equations are related to
projections and limits of the three-dimensional master equations of Hirota and
Miwa, and find that sometimes a singular limit is needed.
