  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 quantum/Moyal deformations of the
self dual membrane in terms of the q-Moyal star product . The q deformations of
the SU(\infty) Nahm equations are studied and explicit solutions are given. The
continuum limit of the q Toda chain equations are obtained furnishing q
deformations of the self dual membrane. Finally, the continuum Moyal-Toda chain
equation is embedded into the SU(\infty) Moyal-Nahm equations, rendering the
relation with the Moyal deformations of the self dual membrane. W_{\infty} and
q-W_{\infty} algebras arise as the symmetry algebras and the role of ( the
recently developed ) quantum Lie algebras associated with quantized universal
enveloping algebras is pointed out pertaining the formulation of a q Toda
theory. We review as well the Weyl-Wigner-Moyal quantization 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.
