  Spatially periodic modulation of the intersite coupling in two-dimensional
(2D) nonlinear lattices modifies the eigenvalue spectrum by opening mini-gaps
in it. This work aims to build stable localized modes in the new bandgaps.
Numerical analysis shows that single-peak and composite two- and four-peak
discrete static solitons and breathers emerge as such modes in certain
parameter areas inside the mini-gaps of the 2D superlattice induced by the
periodic modulation of the intersite coupling along both directions.The
single-peak solitons and four-peak discrete solitons are stable in a part of
their existence domain, while unstable stationary states (in particular,
two-soliton complexes) may readily transform into robust localized breathers.
