*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 consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an*

*integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues*

*and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of*

*the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where*

*the solution is localized.*

*We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an*

*integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues*

*and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of*

*the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where*

*the solution is localized.*

*We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an*

*integral Hartree-type nonlinearity with a Coulomb self-action potential. We obtain asymptotic eigenvalues*

*and asymptotic eigenfunctions near the lower boundaries of spectral clusters formed in a neighborhood of*

*the eigenvalues of the unperturbed operator and construct an asymptotic expansion near a circle where*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

*the solution is localized.*

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.

As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke R-matrix, the elements Tr_R(L^k) (called quantum power sums) are central. Here, L is the generating matrix of this algebra, and Tr_R is the operation of taking the R-trace associated with a given R-matrix. We consider the problem of whether this is true in certain RE-like algebras depending on a spectral parameter. We mainly study algebras similar to those introduced by Reshetikhin and Semenov-Tian-Shansky (we call them algebras of RS type). These algebras are defined using some current R-matrices (i.e., depending on parameters) arising from involutive and Hecke R-matrices by so-called Baxterization. In algebras of RS type. we define quantum power sums and show that the lowest quantum power sum is central iff the value of the “charge” c in its definition takes a critical value. This critical value depends on the birank (m|n) of the initial R-matrix. Moreover, if the birank is equal to (m|m) and the charge c has a critical value, then all quantum power sums are central.

We extend the relation between cluster integrable systems and q-difference equations beyond the Painlev*é* case. We consider the class of hyperelliptic curves where the Newton polygons contain only four boundary points. We present the corresponding cluster integrable Toda systems and identify their discrete automorphisms with certain reductions of the Hirota difference equation. We also construct nonautonomous versions of these equations and find that their solutions are expressed in terms of five-dimensional Nekrasov functions with Chern–Simons contributions, while these equations in the autonomous case are solved in terms of Riemann theta functions.

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 show that both the dKP hierarchy and its strict version can be extended to a wider class of deformations satisfying a larger set of Lax equations. We prove that both extended hierarchies have appropriate linearizations allowing a geometric construction of their solutions.

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.

To obtain a generating function of the most general form for Hurwitz numbers with arbitrary base surfaceand arbitrary ramification profiles, we consider a matrix model constructed according to a graph on anoriented connected surfaceΣwith no boundary. The vertices of this graph, called stars, are small discs,and the graph itself is a clean dessin d’enfants. We insert source matrices in boundary segments of eachdisc. Their product determines the monodromy matrix for a given star, whose spectrum is called the starspectrum. The surfaceΣconsists of glued maps, and each map corresponds to the product of randommatrices and source matrices. Wick pairing corresponds to gluing the surface from the set of maps, and anadditional insertion of a special tau function in the integration measure corresponds to gluing the M ̈obiusbands. We calculate the matrix integral as a Feynman power series in which the star spectrul data playthe role of coupling constants, and the coefficients of this power series are just Hurwitz numbers. Theydetermine the number of coverings ofΣ(or its extensions to a Klein surface obtained by inserting M ̈obiusbands) for any given set of ramification profiles at the vertices of the graph. We focus on a combinatorialdescription of the matrix integral. The Hurwitz number is equal to number of Feynman diagrams of acertain type divided by the order of the automorphism group of the graph

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.