  There exist two versions of the Kadomtsev-Petviashvili equation, related to
the Cartesian and cylindrical geometries of the waves. In this paper we derive
and study a new version, related to the elliptic cylindrical geometry. The
derivation is given in the context of surface waves, but the derived equation
is a universal integrable model applicable to generic weakly-nonlinear
weakly-dispersive waves. We also show that there exist nontrivial
transformations between all three versions of the KP equation associated with
the physical problem formulation, and use them to obtain new classes of
approximate solutions for water waves.
