  We present a systematic method for the construction of discrete Painlev\'e
equations. The method, dubbed `restoration', allows one to obtain all discrete
Painlev\'e equations that share a common autonomous limit, up to homographic
transformations, starting from any one of those limits. As the restoration
process crucially depends on the classification of canonical forms for the
mappings in the QRT family, it can in principle only be applied to mappings
that belong to that family. However, as we show in this paper, it is still
possible to obtain the results of the restoration even when the initial mapping
is not of QRT type (at least for the system at hand, but we believe our
approach to be of much wider applicability). For the equations derived in this
paper we also show how, starting from a form where the independent variable
advances one step at a time, one can obtain versions corresponding to multistep
evolutions.
