The Lax pseudo-differential operator plays a key role in studying the gener
al set of Kadomtsev-Petviashvili equations, although it is normally treated
in a formal way, without worrying about a complete characterization of its
mathematical properties. The aim of this paper is therefore to investigate
the ellipticity condition. For this purpose, after a careful evaluation of
the kernel with the associated symbol, the majorization ensuring elliptici
ty is studied in detail. This leads to non-trivial restrictions on the admi
ssible set of potentials in the Lax operator. When their time evolution is
also considered, the ellipticity conditions turn out to involve derivatives
of the logarithm of the tau -function.