  The exact quantum integrability problem of the membrane is investigated. It
is found that the spherical membrane moving in flat target spacetime
backgrounds is an exact quantum integrable system for a particular class of
solutions of the light-cone gauge equations of motion : a dimensionally-reduced
$SU(\infty)$ Yang-Mills theory to one temporal dimension. Crucial ingredients
are the exact integrability property of the $3D~SU(\infty)$ continuous Toda
theory and its associated dimensionally-reduced $SU(\infty)$ Toda $molecule$
equation whose symmetry algebra is the $U_\infty$ algebra obtained from a
dimensional-reducion of the $W_\infty \oplus {\bar W}_\infty$ algebras that act
naturally on the original $3D$ continuous Toda theory. The $U_\infty$ algebra
is explicitly constructed in terms of exact quantum solutions of the quantized
continuous Toda equation. Highest weight irreducible representations of the
$W_\infty$ algebras are also studied in detail. Continuous and discrete energy
levels are both found in the spectrum . Other relevant topics are discussed in
the conclusion.
