Generalized shift elements and classical [formula omitted]-matrices: Construction and applications

Abstract

A general algorithm is proposed to obtain “shift elements” which are used to construct inhomogeneous Lax operators containing constant terms, and satisfying general linear r-matrix algebra with a non-dynamical classical r-matrix. The proposed construction is illustrated by examples of skew-symmetric rational, non-skew-symmetric “Zp-graded” and “anisotropic irrational” r-matrices for several known classes of Lax operators and integrable systems, such as rational Gaudin systems in an external magnetic field, closed and open Toda chains, and Kovalevskaja and Zhukovski–Volterra integrable systems. New Lax operators and new integrable systems are also described, associated with “anisotropic irrational” r-matrices that generalize Zhukovski–Volterra integrable systems for the Lie algebra cases gl(n) and so(n).