  Starting from a self-dual $SU(\infty)$ Yang-Mills theory in $(2+2)$
dimensions, the Plebanski second heavenly equation is obtained after a suitable
dimensional reduction. The self-dual gravitational background is the cotangent
space of the internal two-dimensional Riemannian surface required in the
formulation of $SU(\infty)$ Yang-Mills theory. A subsequent dimensional
reduction leads to the KP equation in $(1+2)$ dimensions after the relationship
from the Plebanski second heavenly function, $\Omega$, to the KP function, $u$,
is obtained. Also a complexified KP equation is found when a different
dimensional reduction scheme is performed . Such relationship between $\Omega$
and $u$ is based on the correspondence between the $SL(2,R)$ self-duality
conditions in $(3+3)$ dimensions of Das, Khviengia, Sezgin (DKS) and the ones
of $SU(\infty)$ in $(2+2)$ dimensions . The generalization to the
Supersymmetric KP equation should be straightforward by extending the
construction of the bosonic case to the previous Super-Plebanski equation,
found by us in [1], yielding self-dual supergravity backgrounds in terms of the
light-cone chiral superfield, $\Theta$, which is the supersymmetric analog of
$\Omega$. The most important consequence of this Plebanski-KP correspondence is
that $W$ gravity can be seen as the gauge theory of $\phi$-diffeomorphisms in
the space of dimensionally-reduced $D=2+2,~SU^*(\infty)$ Yang-Mills instantons.
These $\phi$ diffeomorphisms preserve a volume-three-form and are, precisely,
the ones which provide the Plebanski-KP correspondence.
