  The group of area preserving diffeomorphisms showed importance in the
problems of self-dual gravity and integrability theory. We discuss how
representations of this infinite-dimensional Lie group can arise in
mathematical physics from pure local considerations. Then using Lie algebra
extensions and cohomology we derive the second Pleba\'{n}ski equation and its
geometry. We do not use K\"ahler or other additional structures but obtain the
equation solely from the geometry of area preserving transformations group. We
conclude that the Pleba\'{n}ski equation is Lie remarkable.
