2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/187127We 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.8 pagesHigh Energy Physics - Theorytext