  We construct new coset realizations of infinite-dimensional linear
$W_3^{\infty}$ symmetry associated with Zamolodchikov's $W_3$ algebra which are
different from the previously explored $sl_3$ Toda realization of
$W_3^{\infty}$. We deduce the Boussinesq and modified Boussinesq equations as
constraints on the geometry of the corresponding coset manifolds.The main
characteristic features of these realizations are:i. Among the coset parameters
there are the space and time coordinates $x$ and $t$ which enter the Boussinesq
equations, all other coset parameters are regarded as fields depending on these
coordinates;ii. The spin 2 and 3 currents of $W_3$ and two spin 1 $U(1)$ Kac-
Moody currents as well as two spin 0 fields related to the $W_3$currents via
Miura maps, come out as the only essential parameters-fields of these cosets.
The remaining coset fields are covariantly expressed through them;iii.The Miura
maps get a new geometric interpretation as $W_3^{\infty}$ covariant constraints
which relate the above fields while passing from one coset manifold to another;
iv. The Boussinesq equation and two kinds of the modified Boussinesq equations
appear geometrically as the dynamical constraints accomplishing $W_3^{\infty}$
covariant reductions of original coset manifolds to their two-dimensional
geodesic submanifolds;v. The zero-curvature representations for these equations
arise automatically as a consequence of the covariant reduction. The approach
proposed could provide a universal geometric description of the relationship
between $W$-type algebras and integrable hierarchies.
