Types of Integrable Systems  

In General > s.a. integrable quantum systems.
* Types: s-integrable, studied by spectral methods; c-integrable, solved by changing variables; Superintegrable, with more than n functionally independent integrals of the motion.
@ 2D: Baleanu & Karasu MPLA(99)gq/00 [Lax, with symmetries]; McLenaghan & Smirnov JMP(00); Pucacco & Rosquist JMP(05).
@ In arbitrary dimensions: Álvarez et al AIP(99)ht; Goliath et al JPA(99)si/98.
@ Hamiltonian models: Magri JMP(78); Calogero & Françoise JMP(96); Mostafazadeh a1401 [differential integrability condition for 2D Hamiltonian systems].
@ Superintegrable: Kalnins et al JPA(01)mp/01, JMP(02)mp/01; Daskaloyannis & Ypsilantis JMP(06)-mp/04 [2D, with integrals quadratic in momenta]; Ballesteros & Herranz JPA(07) [on constant curvature]; Yzaguirre MS-a1209 [geometric structure]; Nucci & Post JPA(12) [and Lie symmetries]; Nikitin JPA(12)-a1205 [new examples]; Gonera & Kaszubska AP(14)-a1311 [on spaces of constant curvature]; > s.a. classical systems; Fock Symmetry.
@ Discrete: Grammaticos et al JPA(01) [integrability]; Kimura et al JPA(02) [and discrete Painlevé]; Quispel et al JPA(05) [duality]; Grammaticos et al JPA(09) [integrability tests].
@ Other types: Sen & Chowdhury JMP(93) [supersymmetric]; Devchand & Ogievetsky ht/94-conf [4D]; Ramani et al JPA(00) [without Painlevé property]; Reshetikhin a1509 [degenerate integrability].

Bi-Hamiltonian and Related Systems > s.a. Bi-Hamiltonian System.
* Bi-Hamiltonian system: A bi-Hamiltonian system is integrable if its Nijenhuis tensor vanishes.
@ Bi-Hamiltonian systems: Smirnov LMP(97) [constructive]; Sergyeyev AAM(04)nl/03 [construction]; Nutku & Pavlov JMP(02) [multiple Lagrangians]; Praught & Smirnov Sigma(05)n.SI [history, Lenard recursion formula]; Bogoyavlenskij DG&A(07) [identity for Schouten tensor]; Gürses et al JMP(09)-a0903 [all dynamical systems on \(\mathbb R\)n are (n−1)-Hamiltonian]; Barnich & Troessaert JMP(09)-a0812 [electromagnetism, linearized gravity and Yang-Mills theory]; Mokhov TMP(11) [non-local, of hydrodynamic type]; Bolsinov & Izosimov CMP(14) [singularities]; > s.a. duality in field theory; integrable quantum systems; types of symplectic structures.
@ Quasi-bi-Hamiltonian systems: Morosi & Tondo JPA(97).

Specific Examples > s.a. types of field theories [integrable]; non-commutative systems; self-dual fields; toda lattice.
* Examples: The Toda lattice and Korteweg-de Vries (KdV) equation; 3-body ones include the Kaluza-Klein two-center problem [@ Cornish & Gibbons CQG(97)gq/96].
@ Calogero-Moser: Calogero in(91), JMP(93); Gonera JMP(98); Bordner et al PTP(98)ht, PTP(99)ht/98, Bordner & Sasaki PTP(99)ht/98; Bordner et al PTP(99) [generalized]; Bordner et al PTP(00).
@ Calogero & Sutherland models: Rühl & Turbiner MPLA(95); Efthimiou & Spector PRA(97)qp; Gurappa & Panigrahi ht/99, PRB(00)ht/99; Forger & Winterhalder ht/99; Jonke & Meljanac PLB(01) [symmetry algebra]; Guhr & Kohler PRE(05)mp/04 [supersymmetric extension]; Sasaki & Takasaki JMP(06) [explicit solutions, any root system]; Polychronakos JPA(06)ht [rev]; in Xu a1205 [algebraic approach].
@ KdV: Nakamura JMP(81) [Bäcklund transformation]; Dimakis & Müller-Hoissen PLA(00)ht [non-commutative]; Kersten & Krasil'shchik n.SI/00 [KdV-mKdV]; Khare & Sukhatme PRL(02)mp/01 [superposition of solutions]; Gieseker JDG(03) [deformation]; Carroll qp/03 [KP/KdV and quantum mechanics]; Hayashi et al PRS(03) [initial-boundary-value problem]; Bracken PhyA(04) [solutions]; Willink a0710-conf [history of Korteweg-de Vries paper]; Rasin & Schiff JPA(09) [discrete, infinitely-many conservation laws]; Lidsey PRD(12)-a1205 [significance to cosmology]; in Xu a1205-ch [algebraic approach]; Zakharov TMP(13) [Cauchy problem, renormalization method]; Karczewska & Rozmej a1901 [higher-order, solutions]; > s.a. heat kernel; solitons.
@ Other examples: Vosmischeva 03 [spaces of constant curvature]; Gadella et al JPA(08)-a0711-conf [some 3D systems]; > s.a. Dimer Models; special potentials [exactly solvable].
> Particle motion in curved spacetimes: see kerr and generalized kerr spacetimes; spinning particles.

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