Reduction in soliton hierarchies and special points of classical r-matrices

We propose the most general approach to construction of the U-V pairs of hierarchies of soliton equations in two dimensions based on the theory of classical non-skew-symmetric r-matrices with spectral parameters and infinite-dimensional Lie algebras. We show that reduction in integrable hierarchies is connected with “special points” of classical r-matrices in which they become singular or degenerated. We prove that “Mikhailov's reduction” or reduction with the help of automorphism is a partial case of our construction. We consider two types of integrable hierarchies and the corresponding soliton equations: the so-called “positive” and “negative flow” equations. We show that the “negative flow” equations can be written in the most general case without a specification of the concrete form of classical r-matrix. They coincide with a generalization of chiral field equation and its different reductions, with a generalization of abelian and non-abelian Toda field equations and new class of integrable equations which we call “double-shift” equations. For the case of “positive” hierarchies and “positive” flows we explicitly write general U-V pairs of “nominative” equations of hierarchy. We consider examples of new equations of such the types and present new soliton equations coinciding with elliptic deformation of dNS equation and its “negative flow” equation and ultimate generalization of the abelian and non-abelian modified Toda field equations.