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.
Categories: hep-th