  With this paper we begin an investigation of difference schemes that possess
Darboux transformations and can be regarded as natural discretizations of
elliptic partial differential equations. We construct, in particular, the
Darboux transformations for the general self adjoint schemes with five and
seven neighbouring points. We also introduce a distinguished discretization of
the two-dimensional stationary Schrodinger equation, described by a 5-point
difference scheme involving two potentials, which admits a Darboux
transformation.
