  The exact quantum integrability aspects of a sector of the membrane is
investigated. It is found that spherical membranes ( in the lightcone gauge)
moving in flat target spacetime backgrounds admit a class of integrable
solutions linked to $SU(\infty)$ SDYM equations ( dimensionally reduced to one
temporal dimension) which, in turn, are related to Plebanski 4D SD
Gravitational equations. A further rotational Killing-symmetry reduction yields
the 3D continuous Toda theory. It is precisely the latter which bears a direct
relationship to non critical $W_\infty$ string theory. The expected critical
dimensions for the ( super) membrane , (D=11) and D=27, are easily obtained.
This suggests that this particular sector of the membrane's spectrum (connected
to the $SU(\infty)$ SDYM equations ) bears a direct connection to a critical
$W_\infty$ string spectrum adjoined to a q=N+1 unitary minimal model of the W_N
algebra in the $N\rightarrow \infty$ limit. Final comments are made about the
connection to Jevicki's observation that the 4D quantum membrane is linked to
dilatonic-self dual gravity plus matter . 2D dilatonic ( super) gravity was
studied by Ikeda and its relation to nonlinear $W_\infty$ algebras from
nonlinear integrable deformations of 4D self dual gravity was studied by the
author.The full $SU(\infty)$ YM theory remains to be explored as well as the
incipient role that noncritical nonlinear $W_\infty$ strings might have in the
full quantization program.
