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## Publications

Publications (123)

We describe the most general ${\rm GL}_{NM}$ classical elliptic finite-dimensional integrable system, which Lax matrix has $n$ simple poles on elliptic curve. For $M=1$ it reproduces the classical inhomogeneous spin chain, for $N=1$ it is the Gaudin type (multispin) extension of the spin Ruijsenaars-Schneider model, and for $n=1$ the model of $M$ i...

We compare the construction of 2D integrable models through two gauge field theories. The first one is the 4D Chern–Simons (4D-CS) theory proposed by Costello and Yamazaki. The second one is the 2D generalization of the Hitchin integrable systems constructed by means of affine Higgs bundles (AHB). We illustrate the latter approach by considering 1...

We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. The Lax pair with spectral parameter is constructed by means of a quantum $R$-matrix satisfying the associative Yang-Baxter...

We describe the most general ${\rm GL}_{NM}$ classical elliptic finite-dimensional integrable system, which Lax matrix has $n$ simple poles on elliptic curve. For $M=1$ it reproduces the classical inhomogeneous spin chain, for $N=1$ it is the Gaudin type (multispin) extension of the spin Ruijsenaars-Schneider model, and for $n=1$ the model of $M$ i...

A bstract
In this study we map the dualities observed in the framework of integrable probabilities into the dualities familiar in a realm of integrable many-body systems. The dualities between the pairs of stochastic processes involve one representative from Macdonald-Schur family, while the second representative is from stochastic higher spin six-...

In this short review we compare constructions of 2d integrable models by means of two gauge field theories. The first one is the 4d Chern-Simons (4d-CS) theory proposed by Costello and Yamazaki. The second one is the 2d generalization of the Hitchin integrable systems constructed by means the Affine Higgs bundles (AHB). We illustrate this approach...

We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Macdonald-Ruijsenaars operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range sp...

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin $R$-matrix in the fundamental representation of ${\rm GL}_M$. In the scalar case $M=1$ these operators are the elliptic Macdonald-Ruijsenaars operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijs...

A bstract
The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin approach. For this purpose we describe double-elliptization of the Cherednik construction. Namely, we derive explicit expression in terms of the Ch...

Приводится обзор и подробное описание $gl_{NM}$ моделей Годена, связанных с голоморфными векторными расслоениями ранга $NM$ и степени $N$ на эллиптической кривой с $n$ проколотыми точками. Определяются их обобщения, построенные с помощью $R$-матриц, удовлетворяющих ассоциативному уравнению Янга-Бакстера. Приводится естественное обобщение полученных...

We introduce a notion of quasi-antisymmetric Higgs G-bundles over curves with marked points. They are endowed with additional structures that replace the parabolic structures at marked points in parabolic Higgs bundles. This means that the coadjoint orbits are attached to the marked points of the curves. The moduli spaces of parabolic Higgs bundles...

In this study we map the dualities observed in the framework of integrable probabilities into the dualities familiar in a realm of integrable many-body systems. The dualities between the pairs of stochastic processes involve one representative from Macdonald-Schur family, while the second representative is from stochastic higher spin six-vertex mod...

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of $L$-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical ellip...

We review and give detailed description for ${\rm gl}_{NM}$ Gaudin models related to holomorphic vector bundles of rank $NM$ and degree $N$ over elliptic curve with $n$ punctures. Then we introduce their generalizations constructed by means of $R$-matrices satisfying the associative Yang-Baxter equation. A natural extension of the obtained models t...

We construct quadratic quantum algebra based on the dynamical RLL-relation for the quantum $R$-matrix related to $SL(NM)$-bundles with nontrivial characteristic class over elliptic curve. This $R$-matrix generalizes simultaneously the elliptic nondynamical Baxter--Belavin and the dynamical Felder $R$-matrices,and the obtained quadratic relations ge...

The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin approach. For this purpose we describe double-elliptization of the Cherednik construction. Namely, we derive explicit expression in terms of the Cherednik op...

We give detailed description for continuous version of the classical IRF-Vertex relation, where on the IRF side we deal with the Calogero-Moser-Sutherland models. Our study is based on constructing modifications of the Higgs bundles of infinite rank over elliptic curve and its degenerations. In this way the previously predicted gauge equivalence be...

We introduce a notion of quasi-antisymmetric Higgs $G$-bundles over curves with marked points. They are endowed with additional structures, which replace the parabolic structures at marked points in the parabolic Higgs bundles. The latter means that the coadjoint orbits are attached to the marked points. The moduli space of parabolic Higgs bundles...

We extend duality between the quantum integrable Gaudin models with boundary and the classical Calogero–Moser systems associated with root systems of classical Lie algebras B N , C N , D N to the case of supersymmetric gl( m | n ) Gaudin models with m + n = 2. Namely, we show that the spectra of quantum Hamiltonians for all such magnets being ident...

A family of integrable $GL(NM)$ models is described. On the one hand it generalizes the classical spin Ruijsenaars--Schneider systems (the case $N=1$), and on the other hand it generalizes the relativistic integrable tops on $GL(N)$ Lie group (the case $M=1$). The described models are obtained by means of the Lax pair with spectral parameter. Equat...

We give detailed description for continuous version of the classical IRF-Vertex relation, where on the IRF side we deal with the Calogero-Moser-Sutherland models. Our study is based on constructing modifications of the Higgs bundles of infinite rank over elliptic curve and its degenerations. In this way the previously predicted gauge equivalence be...

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we r...

Описано семейство интегрируемых $GL(NM)$-моделей, обобщающих, с одной стороны, классические спиновые системы Руйсенарса-Шнайдера (случай $N=1$), а с другой - классические релятивистские интегрируемые волчки на группе Ли $GL(N)$ (случай $M=1$). Модели описываются через пары Лакса со спектральным параметром. Получены уравнения движения. Для построени...

We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application, we construct odd supersymmetric extensions of the elliptic R-matrices, which satisfy the classical and the associative Yang–Baxter equations.

We extend duality between the quantum integrable Gaudin models with boundary and the classical Calogero-Moser systems associated with root systems of classical Lie algebras $B_N$, $C_N$, $D_N$ to the case of supersymmetric ${\rm gl}(m|n)$ Gaudin models with $m+n=2$. Namely, we show that the spectra of quantum Hamiltonians for all such magnets being...

We obtain a determinant representation of normalized scalar products of on-shell and off-shell Bethe vectors in the inhomogeneous 8-vertex model. We consider the case of rational anisotropy parameter and use the generalized algebraic Bethe ansatz approach. Our method is to obtain a system of linear equations for the scalar products, prove its solva...

We obtain a determinant representation of normalized scalar products of on-shell and off-shell Bethe vectors in the inhomogeneous 8-vertex model. We consider the case of rational anisotropy parameter and use the generalized algebraic Bethe ansatz approach. Our method is to obtain a system of linear equations for the scalar products, prove its solva...

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classic...

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians $H_j^{\rm G}$ with particles velocities $\dot q_...

Предложено релятивистское обобщение интегрируемой системы, состоящей из $M$ взаимодействующих эллиптических $gl(N)$-волчков Эйлера-Арнольда. Результат представляет собой эллиптическую интегрируемую систему, которая при ${N=1}$ воспроизводит спиновую эллиптическую $GL(M)$-модель Руйсенарса-Шнайдера, а при $M=1$ - релятивистский эллиптический $GL(N)$...

We propose a relativistic generalization of integrable systems describing M interacting elliptic gl(N) Euler-Arnold tops. The obtained models are elliptic integrable systems that reproduce the spin elliptic GL(M) Ruijsenaars-Schneider model with N = 1 and relativistic integrable GL(N) elliptic tops with M = 1. We construct the Lax pairs with a spec...

We propose relativistic generalization of integrable systems describing $M$ interacting elliptic ${\rm gl}(N)$ tops of the Euler-Arnold type. The obtained models are elliptic integrable systems, which reproduce the spin elliptic ${\rm GL}(M)$ Ruijsenaars-Schneider model for $N=1$ case, while in the $M=1$ case they turn into relativistic integrable...

We study a general ansatz for an odd supersymmetric version of the Kronecker elliptic function, which satisfies the genus one Fay identity. The obtained result is used for construction of the odd supersymmetric analogue for the classical and quantum elliptic $R$-matrices. They are shown to satisfy the classical Yang-Baxter equation and the associat...

We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application we construct an odd supersymmetric extensions of the elliptic $R$-matrices, which satisfy the classical and the associative Yang-Baxter equations.

A bstract
We introduce a family of classical integrable systems describing dynamics of M interacting gl N integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GL N R -matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the sp...

We consider a special class of quantum nondynamical R-matrices in the fundamental representation of GLN with spectral parameter given by trigonometric solutions of the associative Yang–Baxter equation. In the simplest case N=2, these are the well-known 6-vertex R-matrix and its 7-vertex deformation. The R-matrices are used for construction of the c...

In this letter we construct ${\rm GL}_{NM}$-valued dynamical $R$-matrix by means of unitary skew-symmetric solution of the associative Yang-Baxter equation in the fundamental representation of ${\rm GL}_{N}$. In $N=1$ case the obtained answer reproduces the ${\rm GL}_{M}$-valued Felder's $R$-matrix, while in the $M=1$ case it provides the ${\rm GL}...

We introduce a family of classical integrable systems describing dynamics of $M$ interacting ${\rm gl}_N$ integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the ${\rm GL}_N$ $R$-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions...

A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has been established by means of the Lax equation specified on a singular curve and the classical r-matrix depending on the spectral parameter. Generalized Toda systems have also...

We consider a special class of quantum non-dynamical $R$-matrices in the fundamental representation of ${\rm GL}_N$ with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case $N=2$ these are the well-known 6-vertex $R$-matrix and its 7-vertex deformation. The $R$-matrices are used for cons...

We describe the correspondence of the Matsuo–Cherednik type between the quantum n-body Ruijsenaars–Schneider model and the quantum Knizhnik–Zamolodchikov equations related to supergroup GL(N|M). The spectrum of the Ruijsenaars–Schneider Hamiltonians is shown to be independent of the Z2-grading for a fixed value of N+M, so that N+M+1 different qKZ s...

We discuss properties of R-matrix-valued Lax pairs for the elliptic Calogero-Moser model. In particular, we show that the family of Hamiltonians arising from this Lax representation contains only known Hamiltonians and no others. We review the relation of R-matrix-valued Lax pairs to Hitchin systems on bundles with nontrivial characteristic classes...

We describe the correspondence of the Matsuo-Cherednik type between the quantum $n$-body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup $GL(N|M)$. The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the ${\mathbb Z}_2$-grading for a fixed value of $N+M$, so that $N...

In this paper we study factorization formulae for the Lax matrices of the classical Ruijsenaars-Schneider and Calogero-Moser models. We review the already known results and discuss their possible origins. The first origin comes from the IRF-Vertex relations and the properties of the intertwining matrices. The second origin is based on the Schlesing...

In this paper we discuss $R$-matrix-valued Lax pairs for ${\rm sl}_N$ Calogero-Moser model and its relation to integrable quantum long-range spin chain of the Haldane-Shastry-Inozemtsev type. First, we construct the $R$-matrix-valued Lax pairs for the third flow of the classical Calogero-Moser model. Then we notice that the scalar parts (in the aux...

The article is devoted to the study of $R$-matrix valued Lax pairs for $N$-body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum ${\rm GL}_{\tilde N}$ $R$-matrices of Baxter-Belavin type. For $\tilde N=1$ the widely known Krichever's Lax pair with spectral parameter is reproduced. First, we construct the $R$-matrix value...

We consider an analog of the classical Calogero-Sutherland system related to a simple complex Lie group $G$ with two types of interacting spin variables. The system is derived by the symplectic reductions from two free systems (Model I, Model II). We construct the Lax operator and hierarchy of independent integrals of motion. Further, we introduce...

We propose a self-dual form of the ${\rm gl}_N$ Ruijsenaars-Schneider model based on the first order equations in $N+M$ complex variables which include $N$ positions of particles and $M$ dual variables. The latter satisfy equations of motion of the ${\rm gl}_M$ Ruijsenaars-Schneider model. In the elliptic case it holds $M=N$ while for the rational...

We consider the classical Calogero-Sutherland system with two types of interacting spin variables. It can be reduced to the standard Calogero-Sutherland system, when one of the spin variables vanishes. We describe the model in the Hitchin approach and prove complete integrability of the system by constructing the Lax pair and the classical $r$-matr...

We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "off-shell" Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell...

We discuss the Matsuo-Cherednik type correspondence between the quantum Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle quantum Ruijsenaars model, with $n$ being not necessarily equal to $N$. The quasiclassical limit of this construction yields the quantum-classical correspondence between the quantum spin chains and th...

We discuss the correspondence between the Knizhnik-Zamolodchikov equations associated with $GL(N)$ and the $n$-particle quantum Calogero model in the case when $n$ is not necessarily equal to $N$. This can be viewed as a natural "quantization" of the quantum-classical correspondence between quantum Gaudin and classical Calogero models.

We discuss the correspondence between models solved by Bethe ansatz and classical integrable systems of Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric 6-vertex model parametrized by trigonometric (hyperbolic) functions.

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on $R$-matrix description which provides Lax pairs in terms of quantum and classical $R$-matrices. First, we prove that for relativistic (and non-relativistic) tops such Lax pairs with spectral parameter follow from the associ...

In this paper we propose versions of the associative Yang-Baxter equation
which can be applied to quantum dynamical $R$-matrices. As is known quantum
non-dynamical $R$-matrices of Baxter-Belavin type satisfy this equation.
Together with unitarity condition and skew-symmetry it provides the quantum
Yang-Baxter equation and a set of identities useful...

We prove a family of $n$-th order identities for quantum $R$-matrices of
Baxter-Belavin type in fundamental representation. The set of identities
includes the unitarity condition as the simplest one ($n=2$). Our study is
inspired by the fact that the third order identity provides commutativity of
the Knizhnik-Zamolodchikov-Bernard connections. On t...

We extend the quantum-classical duality to the trigonometric (hyperbolic)
case. The duality establishes an explicit relationship between the classical
N-body trigonometric Ruijsenaars-Schneider model and the inhomogeneous twisted
XXZ spin chain on N sites. Similarly to the rational version, the spin chain
data fixes a certain Lagrangian submanifold...

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...

We construct twisted Calogero-Moser (CM) systems with spins as the Hitchin
systems derived from the Higgs bundles over elliptic curves, where transitions
operators are defined by an arbitrary finite order automorphisms of the
underlying Lie algebras. In this way we obtain the spin generalization of the
twisted D'Hoker- Phong and Bordner-Corrigan-Sa...

We consider Yang-Baxter equations arising from its associative analog and
study corresponding exchange relations. They generate finite-dimensional
quantum algebras which have form of coupled ${\rm GL}(N)$ Sklyanin elliptic
algebras. Then we proceed to a natural generalization of the Baxter-Belavin
quantum $R$-matrix to the case ${\rm Mat}(N,\mathbb...

The results obtained in the works supported in part by the Russian Foundation for Basic Research (project 12-02-00594) are briefly reviewed. We mainly focus on interrelations between classical integrable systems, Painlevé-Schlesinger equations and related algebraic structures such as classical and quantum R-matrices. The constructions are explained...

It was shown in our previous paper that quantum ${\rm gl}_N$ $R$-matrices
satisfy noncommutative analogues of the Fay identities in ${\rm gl}_N^{\otimes
3}$. In this paper we extend the list of $R$-matrix valued elliptic function
identities. We propose counterparts of the Fay identities in ${\rm
gl}_N^{\otimes 2}$, the symmetry between the Planck c...

For integrable inhomogeneous supersymmetric spin chains (generalized graded
magnets) constructed employing Y(gl(N|M))-invariant R-matrices in
finite-dimensional representations we introduce the master T-operator which is
a sort of generating function for the family of commuting quantum transfer
matrices. Any eigenvalue of the master T-operator is t...

An old conjecture claims that commuting Hamiltonians of the double-elliptic
integrable system are constructed from the theta-functions associated with
Riemann surfaces from the Seiberg-Witten family, with moduli treated as
dynamical variables and the Seiberg-Witten differential providing the
pre-symplectic structure. We describe a number of theta-c...

In our recent paper we suggested a natural construction of the classical
relativistic integrable tops in terms of the quantum $R$-matrices. Here we
study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$
rational models. The corresponding top is equivalent to the 2-body
Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (C...

We construct special rational glN
Knizhnik-Zamolodchikov-Bernard (KZB) equations with Ñ punctures by deformation of the corresponding quantum glN
rational R-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is τ. At the level of classical mechanics the deformation paramet...

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars an...

We construct a rational integrable system (the rational top) on a coadjoint
orbit of ${\rm SL}_N$ Lie group. It is described by the Lax operator with
spectral parameter and classical non-dynamical skew-symmetric $r$-matrix. In
the case of the orbit of minimal dimension the model is gauge equivalent to the
rational Calogero-Moser (CM) system. To obt...

In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous \( gl \)
n
-invariant XXX spin chain o...

We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ with regular singularities of connections at
marked points. The bundles are classified by their characteristic classes.
These classes are elements of the second cohomology group
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the c...

We describe a construction of elliptic integrable systems based on bundles with nontrivial characteristic classes, especially attending to the bundle-modification procedure, which relates models corresponding to different characteristic classes. We discuss applications and related problems such as the Knizhnik-Zamolodchikov-Bernard equations, class...

We claim that some non-trivial theta-function identities at higher genus can
stand behind the Poisson commutativity of the Hamiltonians of elliptic
integrable systems, which are made from the theta-functions on Jacobians of the
Seiberg-Witten curves. For the case of three-particle systems the genus-2
identities are found and presented in the paper....

Motivated by recent progress in the study of supersymmetric gauge theories we
propose a very compact formulation of spectral duality between XXZ spin chains.
The action of the quantum duality is given by the Fourier transform in the
spectral parameter. We investigate the duality in various limits and, in
particular, prove it for q-->1, i.e. when it...

We propose multidimensional versions of the Painleve VI equation and its
degenerations. These field theories are related to the isomonodromy problems of
flat holomorphic infinite rank bundles over elliptic curves and take the form
of non-autonomous Hamiltonian equations. The modular parameter of curves plays
the role of "time". Reduction of the fie...

In the light of the Quantum Painleve-Calogero Correspondence established in
our previous papers [1,2], we investigate the inverse problem. We imply that
this type of the correspondence (Classical-Quantum Correspondence) holds true
and find out what kind of potentials arise from the compatibility conditions of
the related linear problems. The latter...

We consider topologically non-trivial Higgs G-bundles over Riemann surfaces Σg
with marked points and the corresponding Hitchin systems. We show that if G is not simply-connected, then there exists a finite number of different sectors of the Higgs bundles endowed with the Hitchin Hamiltonians. They correspond to different characteristic classes of...

We discuss quantum dynamical elliptic R-matrices related to arbitrary complex
simple Lie group G. They generalize the known vertex and dynamical R-matrices
and play an intermediate role between these two types. The R-matrices are
defined by the corresponding characteristic classes describing the underlying
vector bundles. The latter are related to...

This paper is a continuation of our paper Levin et al. [1]. We consider Modified Calogero–Moser (CM) systems corresponding to the Higgs bundles with an arbitrary characteristic class over elliptic curves. These systems are generalization of the classical Calogero–Moser systems with spin related to simple Lie groups and contain CM subsystems related...

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB)
equations related to the WZW-theory corresponding to the adjoint $G$-bundles of
different topological types over complex curves $\Sigma_{g,n}$ of genus $g$
with $n$ marked points. The bundles are defined by their characteristic classes
- elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G...

This paper is a continuation of our previous paper where the Painlevé-Calogero correspondence has been extended to auxiliary linear problems associated with Painlevé equations. We have proved, for the first five equations from the Painlevé list, that one of the linear problems can be recast in the form of the non-stationary Schrödinger equation who...

In our recent paper we described relationships between integrable systems
inspired by the AGT conjecture. On the gauge theory side an integrable spin
chain naturally emerges while on the conformal field theory side one obtains
some special reduced Gaudin model. Two types of integrable systems were shown
to be related by the spectral duality. In thi...