[en] We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified (or additional) terms because of a B type reduction condition of this integrable hierarchy. Further we show that the D type Drinfeld-Sokolov hierarchy, which is a reduction of the two-component BKP hierarchy, possess a complete Block type additional symmetry algebra. That D type Drinfeld-Sokolov hierarchy has a similar algebraic structure as the bigraded Toda hierarchy which is a differential-discrete integrable system

[en] A general elliptic N × N matrix Lax scheme is presented, leading to two classes of elliptic lattice systems, one which we interpret as the higher-rank analogue of the Landau–Lifschitz equations, while the other class we characterize as the higher-rank analogue of the lattice Krichever–Novikov equation (or Adler's lattice). We present the general scheme, but focus mainly on the latter type of models. In the case N = 2 we obtain a novel Lax representation of Adler's elliptic lattice equation in its so-called 3-leg form. The case of rank N = 3 is analyzed using Cayley's hyperdeterminant of format 2×2×2, yielding a multi-component system of coupled 3-leg quad-equations. (paper)

[en] Three types of operator representations are found for the Lax equations with a self-consistent source. Two of them are shown to be equivalent. 5 refs

[en] The equivalence between bi-Hamiltonian formulation and Lax representation of dispersionless integrable hierarchies associated with two-primary models of Frobenius manifolds is explicitly verified

[en] All q-Painleve equations which are obtained from the q-analog of the sixth Painleve equation are expressed in a Lax formalism. They are characterized by the data of the associated linear q-difference equations. The degeneration pattern of the q-Painleve equations is also presented

[en] We propose the Lax operators for N=2 supersymmetric matrix generalization of the bosonic (1, s)-KdV hierarchies. The simplest examples - the N=2 supersymmetric a=4 KdV and a=5/2 Boussinesq hierarchies - are discussed in detail

[en] We introduce integrable multicomponent non-commutative lattice systems, which can be considered as analogues of the modified Gel’fand–Dikii hierarchy. We present the corresponding systems of Lax pairs and show directly the multidimensional consistency of these Gel’fand–Dikii-type equations. We demonstrate how the systems can be obtained as periodic reductions of the non-commutative lattice Kadomtsev–Petviashvilii hierarchy. The geometric description of the hierarchy in terms of Desargues maps helps to derive a non-isospectral generalization of the non-commutative lattice-modified Gel’fand–Dikii systems. We show also how arbitrary functions of single arguments appear naturally in our approach when making commutative reductions, which we illustrate on the non-isospectral non-autonomous versions of the lattice-modified Korteweg–de Vries and Boussinesq systems. (paper)

No abstract available

[en] A new N=1 supersymmetric Harry Dym equation is constructed by applying supersymmetric reciprocal transformation to a trivial supersymmetric Harry Dym equation, and its recursion operator and Lax formulation are also obtained. Within the framework of symmetry approach, a class of 3rd order supersymmetric equations of Harry Dym type are considered. In addition to five known integrable equations, a new supersymmetric equation, admitting 5th order generalized symmetry, is shown to be linearizable through supersymmetric reciprocal transformation. Furthermore, its Lax representation and recursion operator are given so that the integrability of this new equation is confirmed. -- Highlights: ► A new supersymmetric Harry Dym equation is constructed through supersymmetric reciprocal transformations. ► The recursion operator and Lax formulation are established for the new supersymmetric Harry Dym equation. ► A supersymmetric equation of Harry Dym type is shown to be linearized through supersymmetric reciprocal transformation.

No abstract available