D=11 Supermembrane Instantons, W(infty) Strings and the Super Toda Molecule

Exact instanton solutions to $D=11$ spherical supermembranes moving in flat target spacetime backgrounds are construted. Our starting point is Super Yang-Mills theories, based on the infinite dimensional $SU(\infty)$ group, dimensionally reduced to one time dimension. In this fashion the super-Toda molecule equation is recovered preserving only one supersymmetry out of the $N=16$ that one would have obtained otherwise. It is conjectured that the expected critical target spacetime dimensions for the (super) membrane, ($D=11$) $D=27$ is closely related to that of the $noncritical$ (super) $W_{\infty}$ strings. A BRST analysis of these symmetries should yield information about the quantum consistency of the ($D=11$) $D=27$ dimensional (super) membrane. Comments on the role that Skyrmions might play in the two types of Spinning- Membrane actions construted so far is presented at the conclusion. Finally, the importance that integrability on light-lines in complex superspaces has in other types of solutions is emphasized.