Methods of geometry of differential equations in analysis of the integrable field theory models

In 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 $T_t=T_{xxx}+T{T}_x$ 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 Schr"odinger equation type systems ${\mathrm{\Psi }}_t=i{\mathrm{\Psi }}_{xx}+if(\mathrm{\mid }\mathrm{\Psi }\mathrm{\mid })\mathrm{\Psi }$ (multi-soliton complexes) are described.