  The reduction by restricting the spectral parameters $k$ and $k'$ on a
generic algebraic curve of degree $\mathcal{N}$ is performed for the discrete
AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete
integrable systems possessing a more general solution structure arise from the
reduction, and in each case a unified formula for generic positive integer
$\mathcal{N}\geq 2$ is given to express the corresponding reduced integrable
lattice equations. The obtained extended two-dimensional lattice models give
rise to many important integrable partial difference equations as special
degenerations. Some new integrable lattice models such as the discrete
Sawada--Kotera, Kaup--Kupershmidt and Hirota--Satsuma equations in extended
form are given as examples within the framework.
