[en] The theory of Hamiltonian and quasi-Hamiltonian systems with respect to Nambu-Poisson structures is studied. It is proved that if a dynamical system is endowed with certain properties related to the theory of symmetries then it can be considered as a quasi-Hamiltonian (or Hamiltonian) system with respect to an appropriate Nambu-Poisson structure. Several examples of this construction are presented. These examples are related to integrability and also to superintegrability

[en] In this paper, we construct the addition formulae for several integrable hierarchies, including the discrete KP, the q-deformed KP, the two-component BKP and the D type Drinfeld–Sokolov hierarchies. With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies, we prove that these addition formulae are equivalent to these hierarchies. These studies show that the addition formula in the research of the integrable systems has good universality. (paper)

[en] The formal series symmetry approach (FSSA), a quite powerful and straightforward method to establish infinitely many generalized symmetries of classical integrable systems, has been successfully extended in the supersymmetric framework to explore series of infinitely many generalized symmetries for supersymmetric systems. Taking the N = 1 supersymmetric Boiti–Leon–Manna–Pempinelli system as a concrete example, it is shown that the application of the extended FSSA to this supersymmetric system leads to a set of infinitely many generalized symmetries with an arbitrary function f (t). Some interesting special cases of symmetry algebras are presented, including a limit case f (t) = 1 related to the commutativity of higher order generalized symmetries. (paper)

[en] The Abel-Plana formula (APF) is a widely used tool for calculations in Casimir-type problems. In this paper, we present a particular explicit modification of the generalized APF for the functions with non-integrable branch-point singularities

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No abstract available

[en] We examine a family of difference equations, known as the Lyness mappings, from the point of view of integrability. We show that the mappings satisfy two different integrability criteria and are thus good integrability candidates. We introduce an ansatz which reduces the mappings to bilinear form and we show that the equations obtained are just reductions of the Hirota-Miwa equation which establishes the integrable character of the Lyness mappings. Finally, we discuss the construction of explicit invariants for some instances of the mapping.

[en] A new loop algebra X is constructed, which is devoted to working out the well-known Blaszak-Marciniak lattice equations hierarchy. Then an extended algebraic system X∼ of X is presented, from which the integrable coupling system of the Blaszak-Marciniak lattice equations are obtained

[en] We demonstrate that statistics of certain classes of set partitions are described by generating functions related to the Burgers, Ibragimov–Shabat and Korteweg–de Vries integrable hierarchies. (paper)

[en] It is shown that the Kronecker product can be applied to constructing new discrete integrable coupling system of soliton equation hierarchy in this paper. A direct application to the fractional cubic Volterra lattice spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is also indicated that the study of discrete integrable couplings by using the Kronecker product is an efficient and straightforward method. This method can be used generally