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 3 x 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 Schrodinger equation type systems Ψt = iΨxx + if(|Ψ|) Ψ (multi-soliton complexes) are described.