  Since the lightcone self dual spherical membrane, moving in flat target
backgrounds, has a direct correspondence with the $SU(\infty)$ Nahm equations
and the continuous Toda theory, we construct the Moyal deformations of the self
dual membrane in terms of the Moyal deformations of the continuous Toda theory.
This is performed by using the Weyl-Wigner-Moyal quantization technique of the
3D continuous Toda field equation, and its associated 2D continuous Toda
molecule, based on Moyal deformations of rotational Killing symmetry reductions
of Plebanski first heavenly equation associated with 4D Self Dual Gravity.
