The aim of this paper is two-fold. First, we define symplectic maps between Hitchin systems related to holomorphic bundles of different degrees. We call these maps the Symplectic Hecke Correspondence (SHC) of the corresponding Higgs bundles. They are constructed by means of the Hecke correspondence of the underlying holomorphic bundles. SHC allows to construct Bäcklund transformations in the Hitchin systems defined over Riemann curves with marked points. We apply the general scheme to the elliptic Calogero-Moser (CM) system and construct SHC to an integrable SL(N, C) Euler-Arnold top (the elliptic SL(N, C)-rotator). Next, we propose a generalization of the Hitchin approach to 2d integrable theories related to the Higgs bundles of infinite rank. The main example is an integrable two-dimensional version of the two-body elliptic CM system. The previous construction allows to define SHC between the two-dimensional elliptic CM system and the Landau-Lifshitz equation.

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We show that the Painlevé VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in C 3 and include the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions. June 4, 2018 PVI was discovered by B.Gambier [10] in 1906. He accomplished the Painlevé classification program of the second order differential equations whose solutions have not movable critical points. This equation and its degenerations P V − P I have a lot of applications in Theoretical and Mathematical Physics. (see, for example [41]).We prove here that PVI can be write down in a very simple form as ODE with a quadratic non-linearity. It is a non-autonomous version of the SL(2, C) Zhukovsky-Volterra gyrostat (ZVG) [42,39]. The ZVG generalizes the standard Euler top in the space C 3 by adding an external constant rotator momentum. The ZVG equation describes the evolution of the momentum vector S = (S 1 , S 2 , S 3 ) lying on a SL(2, C) coadjoint orbit. We consider Non-Autonomous Zhukovsky-Volterra gyrostat (NAZVG)where J(τ ) · S = (J 1 S 1 , J 2 S 2 , J 3 S 3 ). Three additional constants ν ′ = (ν ′ 1 , ν ′ 2 , ν ′ 3 ) form the gyrostat momentum vector, and the vector J = {J α (τ )}, (α = 1, 2, 3) is the inverse inertia vector CRDF RM1-2545. The work of A.Z. was also partially supported by the grant MK-2059MK- .2005.2. We are grateful for hospitality to the Max Planck Institute of Mathematics at Bonn where two of us (A.L. and M.O.) were staying during preparation of this paper. We would like to thank a referee for valuable remarks, that allows us to improve the paper.

We investigate the classical limit of the Knizhnik-Zamolodchikov-Bernard equations, considered as a system of non-stationar Schröodinger equations on singular curves, where times are the moduli of curves. It has a form of reduced nonautonomous hamiltonian systems which include as particular examples the Schlesinger equations, Painlevé VI equation and their generalizations. In general case, they are defined as hierarchies of isomonodromic deformations (HID) with respect to changing the moduli of underling curves. HID are accompanying with the Whitham hierarchies. The phase space of HID is the space of flat connections of G bundles with some additional data in the marked points. HID can be derived from some free field theory by the hamiltonian reduction under the action of the gauge symmetries and subsequent factorization with respect to diffeomorphisms of curve. This approach allows to define the Lax equations associated with HID and the linear system whose isomonodromic deformations are provided by HID. In addition, it leads to description of solutions of HID by the projection method. In some special limit HID convert into the Hitchin systems. In particular, for SL(N, C) bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero N -body system.1 Quantization of isomonodromic deformations on rational and elliptic curves and their relations to KZB was considered in [17,18,19] 2 Flat bundles over singular curvesWe will describe the general setup more or less naively. We don't consider in this section the mechanism of symplectic reduction in detail and postpone it on Sect.4 We will consider three cases: 1) smooth proper (compact) algebraic curves; 2) smooth proper algebraic curves with punctures; 3) proper algebraic curves with nodal singularities (double points).Let G be a semisimple group and V be its exact representation; f.e. G = SL(N, C) and V is a N-dimensional vector space with a volume form. Smooth curves.Let S be a smooth oriented compact surface of genus g. Let us consider the moduli space F Bun S,G of flat V -bundles on S. This space can be considered as the quotient of

We construct, for a field

We consider topologically non-trivial Higgs bundles over elliptic curves with marked points and construct corresponding integrable systems. In the case of one marked point we call them the modified Calogero-Moser systems (MCM systems). Their phase space has the same dimension as the phase space of the standard CM systems with spin, but less number of particles and greater number of spin variables. Topology of the holomorphic bundles are defined by their characteristic classes. Such bundles occur if G has a non-trivial center, i.e. classical simply-connected groups, E 6 and E 7 . We define the conformal version CG of G -an analog of GL(N) for SL(N), and relate the characteristic classes with degrees of CGbundles. Starting with these bundles we construct Lax operators, quadratic Hamiltonians, define the phase spaces and the Poisson structure using dynamical r-matrices. To describe the systems we use a special basis in the Lie algebras that generalizes the basis of t'Hooft matrices for sl(N). We find that the MCM systems contain the standard CM systems related to some (unbroken) subalgebras. The configuration space of the CM particles is the moduli space of the holomorphic bundles with non-trivial characteristic classes.

We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N, C) for any N , whose inertia ellipsiod is related to a choice of an elliptic curve. Its bihamiltonian structure is provided by the compatible linear and quadratic Poisson brackets, both of which are governed by the Belavin-Drinfeld classical elliptic r-matrix. We also generalize this bihamiltonian construction of integrable Euler-Arnold tops to several infinitedimensional groups, appearing as certain large N limits of GL(N, C). These are the group of a non-commutative torus (NCT) and the group of symplectomorphisms SDif f (T 2 ) of the two-dimensional torus. The elliptic rotator on symplectomorphisms gives an elliptic version of an ideal 2D hydrodynamics, which turns out to be an integrable system. In particular, we define the quadratic Poisson algebra on the space of Hamiltonians on T 2 depending on two irrational numbers. In conclusion, we quantize the infinite-dimensional quadratic Poisson algebra in a fashion similar to the corresponding finite-dimensional case.

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