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

SDiff(2) and uniqueness of the Pleba\'{n}ski equation