“A`−1” stands for the A`−1 root system that underlies this model. Similarly, an elliptic Calogero-Moser system can be defined for each irreducible (but not necessary reduced) root system. Furthermore, for non-simply laced root systems, a kind of variants called “twisted model” and “extended twisted models” are also known. Those elliptic Calogero-Moser systems are known to possess an isospectral...

A systematic approach to Liouville integrable defects is proposed, based on an underlying Poisson algebraic structure. The non-linear Schrödinger model in the presence of a single particle-like defect is investigated through this algebraic approach. Local integrals of motions are constructed as well as the time components of the corresponding Lax pairs. Continuity conditions imposed upon the ti...

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P Es) is reviewed. The method assumes that the P Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for ...

In this paper we describe integrable generalizations of the classical Steklov– Lyapunov systems, which are defined on a certain product so(m) × so(m), as well as the structure of rank r coadjoint orbits in so(m) × so(m). We show that the restriction of these systems onto some subvarieties of the orbits written in new matrix variables admits a new r × r matrix Lax representation in a generalized...

Lax pair and Hamiltonian formulations of the N = 4 supersymmetric Toda chain (KdV) hierarchy in N = 4 superspace are proposed. The general formulae for the infinite tower of its bosonic flows in terms of the Lax operator in N = 4 superspace are derived. A new N = 4 superfield basis in which the flows are local is constructed and its embedding in the N = 4 O(4) superconformal supercurrent is est...

A non-isospectral linear problem for an integrable 2+1 generalization of the non Schr\"odinger equation, which includes dispersive terms third and fourth order, is presented. The classical symmetries Lax pair related reductions are carefully studied. We obtain several that yield in some cases problems 1+1 dimensions.

A chain of one-dimensional Schrödinger operators is called a “dressing chain” if they are connected by successive Darboux transformations. Particularly interesting are periodic dressing chains; they include finite-band operators and Painlevé equations as a special case. We investigate the Hamiltonian structure of these nonlinear lattices using V. Adler’s 2× 2 Lax pair. The Lax equation and the ...

The tri-Hamiltonian method is applied to the Merola-Ragnisco-Tu equation. This enables us construct a new integrable system, whose continuum limit related AKNS system. Moreover, system proved have linear spectral problems (Lax pair), bi-Hamiltonian structure and Darboux-Bäcklund transformation. Through transformation, we some exact solutions for