| Introduction | p. 1 |
| Integrable dynamical systems | p. 5 |
| Introduction | p. 5 |
| The Liouville theorem | p. 7 |
| Action-angle variables | p. 10 |
| Lax pairs | p. 11 |
| Existence of an r-matrix | p. 13 |
| Commuting flows | p. 17 |
| The Kepler problem | p. 17 |
| The Euler top | p. 19 |
| The Lagrange top | p. 20 |
| The Kowalevski top | p. 22 |
| The Neumann model | p. 23 |
| Geodesics on an ellipsoid | p. 25 |
| Separation of variables in the Neumann model | p. 27 |
| Synopsis of integrable systems | p. 32 |
| Examples of Lax pairs with spectral parameter | p. 33 |
| The Zakharov-Shabat construction | p. 35 |
| Coadjoint orbits and Hamiltonian formalism | p. 41 |
| Elementary flows and wave function | p. 49 |
| Factorization problem | p. 54 |
| Tau-functions | p. 59 |
| Integrable field theories and monodromy matrix | p. 62 |
| Abelianization | p. 65 |
| Poisson brackets of the monodromy matrix | p. 72 |
| The group of dressing transformations | p. 74 |
| Soliton solutions | p. 79 |
| Algebraic methods | p. 86 |
| The classical and modified Yang-Baxter equations | p. 86 |
| Algebraic meaning of the classical Yang-Baxter equations | p. 89 |
| Adler-Kostant-Symes scheme | p. 92 |
| Construction of integrable systems | p. 94 |
| Solving by factorization | p. 96 |
| The open Toda chain | p. 97 |
| The r-matrix of the Toda models | p. 100 |
| Solution of the open Toda chain | p. 105 |
| Toda system and Hamiltonian reduction | p. 109 |
| The Lax pair of the Kowalevski top | p. 115 |
| Analytical methods | p. 124 |
| The spectral curve | p. 125 |
| The eigenvector bundle | p. 130 |
| The adjoint linear system | p. 138 |
| Time evolution | p. 142 |
| Theta-functions formulae | p. 145 |
| Baker-Akhiezer functions | p. 149 |
| Linearization and the factorization problem | p. 153 |
| Tau-functions | p. 154 |
| Symplectic form | p. 156 |
| Separation of variables and the spectral curve | p. 162 |
| Action-angle variables | p. 164 |
| Riemann surfaces and integrability | p. 167 |
| The Kowalevski top | p. 169 |
| Infinite-dimensional systems | p. 175 |
| The closed Toda chain | p. 178 |
| The model | p. 178 |
| The spectral curve | p. 181 |
| The eigenvectors | p. 182 |
| Reconstruction formula | p. 184 |
| Symplectic structure | p. 191 |
| The Sklyanin approach | p. 193 |
| The Poisson brackets | p. 196 |
| Reality conditions | p. 200 |
| The Calogero-Moser model | p. 206 |
| The spin Calogero-Moser model | p. 206 |
| Lax pair | p. 208 |
| The r-matrix | p. 210 |
| The scalar Calogero-Moser model | p. 214 |
| The spectral curve | p. 216 |
| The eigenvector bundle | p. 218 |
| Time evolution | p. 220 |
| Reconstruction formulae | p. 221 |
| Symplectic structure | p. 223 |
| Poles systems and double-Bloch condition | p. 226 |
| Hitchin systems | p. 232 |
| Examples of Hitchin systems | p. 239 |
| The trigonometric Calogero-Moser model | p. 244 |
| Isomonodromic deformations | p. 249 |
| Introduction | p. 249 |
| Monodromy data | p. 251 |
| Isomonodromy and the Riemann-Hilbert problem | p. 262 |
| Isomonodromic deformations | p. 264 |
| Schlesinger transformations | p. 270 |
| Tau-functions | p. 272 |
| Ricatti equation | p. 277 |
| Sato's formula | p. 278 |
| The Hirota equations | p. 280 |
| Tau-functions and theta-functions | p. 282 |
| The Painleve equations | p. 290 |
| Grassmannian and integrable hierarchies | p. 299 |
| Introduction | p. 299 |
| Fermions and GL [infinity] | p. 303 |
| Boson-fermion correspondence | p. 308 |
| Tau-functions and Hirota bilinear identities | p. 311 |
| The KP hierarchy and its soliton solutions | p. 314 |
| Fermions and Grassmannians | p. 316 |
| Schur polynomials | p. 322 |
| From fermions to pseudo-differential operators | p. 328 |
| The Segal-Wilson approach | p. 331 |
| The KP hierarchy | p. 338 |
| The algebra of pseudo-differential operators | p. 338 |
| The KP hierarchy | p. 341 |
| The Baker-Akhiezer function of KP | p. 344 |
| Algebro-geometric solutions of KP | p. 348 |
| The tau-function of KP | p. 352 |
| The generalized KdV equations | p. 355 |
| KdV Hamiltonian structures | p. 359 |
| Bihamiltonian structure | p. 363 |
| The Drinfeld-Sokolov reduction | p. 364 |
| Whitham equations | p. 370 |
| Solution of the Whitham equations | p. 379 |
| The KdV hierarchy | p. 382 |
| The KdV equation | p. 382 |
| The KdV hierarchy | p. 386 |
| Hamiltonian structures and Virasoro algebra | p. 392 |
| Soliton solutions | p. 394 |
| Algebro-geometric solutions | p. 398 |
| Finite-zone solutions | p. 408 |
| Action-angle variables | p. 414 |
| Analytical description of solitons | p. 419 |
| Local fields | p. 425 |
| Whitham's equations | p. 433 |
| The Toda field theories | p. 443 |
| The Liouville equation | p. 443 |
| The Toda systems and their zero-curvature representations | p. 445 |
| Solution of the Toda field equations | p. 447 |
| Hamiltonian formalism | p. 454 |
| Conformal structure | p. 456 |
| Dressing transformations | p. 463 |
| The affine sinh-Gordon model | p. 467 |
| Dressing transformations and soliton solutions | p. 471 |
| N-soliton dynamics | p. 474 |
| Finite-zone solutions | p. 481 |
| Classical inverse scattering method | p. 486 |
| The sine-Gordon equation | p. 486 |
| The Jost solutions | p. 487 |
| Inverse scattering as a Riemann-Hilbert problem | p. 496 |
| Time evolution of the scattering data | p. 497 |
| The Gelfand-Levitan-Marchenko equation | p. 498 |
| Soliton solutions | p. 502 |
| Poisson brackets of the scattering data | p. 505 |
| Action-angle variables | p. 510 |
| Symplectic geometry | p. 516 |
| Poisson manifolds and symplectic manifolds | p. 516 |
| Coadjoint orbits | p. 522 |
| Symmetries and Hamiltonian reduction | p. 525 |
| The case M = T*G | p. 532 |
| Poisson-Lie groups | p. 534 |
| Action of a Poisson-Lie group on a symplectic manifold | p. 538 |
| The groups G and G* | p. 540 |
| The group of dressing transformations | p. 542 |
| Riemann surfaces | p. 545 |
| Smooth algebraic curves | p. 545 |
| Hyperelliptic curves | p. 547 |
| The Riemann-Hurwitz formula | p. 549 |
| The field of meromorphic functions of a Riemann surface | p. 549 |
| Line bundles on a Riemann surface | p. 551 |
| Divisors | p. 553 |
| Chern class | p. 554 |
| Serre duality | p. 554 |
| The Riemann-Roch theorem | p. 556 |
| Abelian differentials | p. 559 |
| Riemann bilinear identities | p. 560 |
| Jacobi variety | p. 562 |
| Theta-functions | p. 563 |
| The genus 1 case | p. 567 |
| The Riemann-Hilbert factorization problem | p. 568 |
| Lie algebras | p. 571 |
| Lie groups and Lie algebras | p. 571 |
| Semi-simple Lie algebras | p. 574 |
| Linear representations | p. 580 |
| Real Lie algebras | p. 583 |
| Affine Kac-Moody algebras | p. 587 |
| Vertex operator representations | p. 592 |
| Index | p. 599 |
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