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

In this paper, we investigate algebraic and geometric properties of hyperbolic Toda equations $u_{xy}=\exp(Ku)$ associated with nondegenerate symmetrizable matrices $K$. A hierarchy of analogues of the potential modified Korteweg"– de Vries equation $u_t=u_{xxx}+u_x^3$ is constructed and its relationship 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 $\Psi_t=\boldsymbol i\Psi_{xx}+\boldsymbol if(|\Psi|)\Psi$ (multi-soliton complexes) are described.