  We deduce the $sl_{3}$ Toda realization of classical $W_3$ symmetry on two
scalar fields in a geometric way, proceeding from a nonlinear realization of
some associate higher-spin symmetry $W_{3}^{\infty}$. The Toda equations are
recognized as the constraints singling out a two-dimensional fully geodesic
subspace in the initial coset space of $W_{3}^{\infty}$. The proposed geometric
approach can be extended to other nonlinear algebras and integrable systems.
