*We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric*(*hyperbolic*) *functions*.

We consider the eigenvalue problem for the Hartree operator with a small parameter multiplying the

nonlinearity. We obtain asymptotic eigenvalues and asymptotic eigenfunctions near the upper boundaries

of spectral clusters formed near the energy levels of the unperturbed operator. Near the circle where

the solution is localized, the leading term of the expansion is a solution of the two-dimensional oscillator problem.

We describe the properties of the Cauchy–Jost (also known as Cauchy–Baker–Akhiezer) function of the Kadomtsev–Petviashvili-II equation. Using the ∂-method, we show that for this function, all equations of the Kadomtsev–Petviashvili-II hierarchy are given in a compact and explicit form, including equations for the Cauchy–Jost function itself, time evolutions of the Jost solutions, and evolutions of the potential of the heat equation.

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.

We investigate the connection between the models of topological conformal theory and noncritical string theory with Saito Frobenius manifolds. For this, we propose a new direct way to calculate the flat coordinates using the integral representation for solutions of the Gauss–Manin system connected with a given Saito Frobenius manifold. We present explicit calculations in the case of a singularity of type *A**n*. We also discuss a possible generalization of our proposed approach to *SU*(*N*)*k*/(*SU*(*N*)*k*+1 × *U*(1)) Kazama–Suzuki theories. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the Frobenius manifold structure to emerge on its deformation manifold. This fact allows using the Dijkgraaf–Verlinde–Verlinde approach to solve similar Kazama–Suzuki models.

We consider the theory of multicomponent free massless fermions in two dimensions and use it to construct representations of *W*-algebras at integer Virasoro central charges. We define the vertex operators in this theory in terms of solutions of the corresponding isomonodromy problem. We use this construction to obtain some new insights into tau functions of the multicomponent Toda-type hierarchies for the class of solutions given by the isomonodromy vertex operators and to obtain a useful representation for tau functions of isomonodromic deformations.

By generalized Yangians, we mean Yangian-like algebras of two different classes. One class comprises the previously introduced so-called braided Yangians. Braided Yangians have properties similar to those of the reflection equation algebra. Generalized Yangians of the second class, RT T -type Yangians, are defined by the same formulas as the usual Yangians but with other quantum R-matrices. If such an R-matrix is the simplest trigonometric R-matrix, then the corresponding RT T -type Yangian is called a q-Yangian. We claim that each generalized Yangian is a deformation of the commutative algebra Sym(gl(m)[t−1]) if the corresponding R-matrix is a deformation of the flip operator. We give the explicit form of the corresponding Poisson brackets.

We construct twisted Calogero–Moser systems with spins as Hitchin systems derived from the Higgs bundles over elliptic curves, where the transition operators are defined by arbitrary finite-order automorphisms of the underlying Lie algebras. We thus obtain a spin generalization of the twisted D’Hoker–Phong and Bordner–Corrigan–Sasaki–Takasaki systems. In addition, we construct the corresponding twisted classical dynamical r-matrices and the Knizhnik–Zamolodchikov–Bernard equations related to the automorphisms of Lie algebras.

We study a family of nonautonomous generalized Liénard-type equations. We consider the equivalence problem via the generalized Sundman transformations between this family of equations and type-I Painlevé–Gambier equations. As a result, we find four criteria of equivalence, which give four integrable families of Liénard-type equations. We demonstrate that these criteria can be used to construct general traveling-wave and stationary solutions of certain classes of diffusion–convection equations. We also illustrate our results with several other examples of integrable nonautonomous Liénard-type equations.

Let E 0 be a holomorphic vector bundle over P1(C) and †0 be a meromorphic connection of E 0. We introduce the notion of an integrable connection that describes the movement of the poles of †0 in the complex plane with integrability preserved. We show the that such a deformation exists under sufficiently weak conditions on the deformation space. We also show that if the vector bundle E0 is trivial, then the solutions of the corresponding nonlinear equations extend meromorphically to the deformation space.

Quantum elliptic R-matrices satisfy the associative Yang-Baxter equation in Mat(N)⊗2, which can be regarded as a noncommutative analogue of the Fay identity for the scalar Kronecker function. We present a broader list of R-matrix-valued identities for elliptic functions. In particular, we propose an analogue of the Fay identities in Mat(N)⊗2. As an application, we use the ℤN×ℤN elliptic R-matrix to construct R-matrix-valued 2N2×2N2 Lax pairs for the Painlevé VI equation (in the elliptic form) with four free constants. More precisely, the case with four free constants corresponds to odd N, and even N corresponds to the case with a single constant in the equation. © 2015, Pleiades Publishing, Ltd.