  The paper presents an approach to derive finite genus solutions to the
lattice potential Kadomtsev-Petviashvili (lpKP) equation introduced by F.W.
Nijhoff, et al. This equation is rederived from compatible conditions of three
replicas of the discrete ZS-AKNS spectral problem, which is a Darboux
transformation of the continuous ZS-AKNS spectral problem. With the help of
these links and by means of the so called nonlinearization technique and
Liouville platform, finite genus solutions of the lpKP equation are derived.
Semi-discrete potential KP equations with one and two discrete arguments,
respectively, are also discussed.
