2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/80757In this paper, we investigate the algebraic and geometric properties of the hyperbolic Toda equations $u_{xy}=\exp(Ku)$ associated with nondegenerate symmetrizable matrices $K$. A hierarchy of analogs to the potential modified Korteweg-de Vries equation $u_t=u_{xxx}+u_x^3$ is constructed, and its relation with the hierarchy for the Korteweg-de Vries equation $T_t=T_{xxx}+TT_x$ is established. Group-theoretic structures for the dispersionless (2+1)-dimensional Toda equation $u_{xy}=\exp(-u_{zz})$ are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems $Ψ_t = iΨ_{xx} + i f(|Ψ|) Ψ$ (multi-soliton complexes) are described.LaTeX2e, 134 pages, no figures, uses diagrams.tex. Submitted to: Fundamental'naya i Prikladnaya Matematika/J. Math. SciExactly Solvable and Integrable Systemstext