  We study the existence of log-canonical Poisson structures that are preserved
by difference equations of special form. We also study the inverse problem,
given a log-canonical Poisson structure to find a difference equation
preserving this structure. We give examples of quadratic Poisson structures
that arise for the Kadomtsev-Petviashvili (KP) type maps which follow from a
travelling-wave reduction of the corresponding integrable partial difference
equation.
