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{{Short description|Property of certain dynamical systems}}

In mathematics, '''integrability''' is a property of certain [[dynamical system]]s. While there are several distinct formal definitions, informally speaking, an '''integrable system''' is a dynamical system with sufficiently many [[conserved quantity|conserved quantities]], or [[first integral]]s,  that its motion is confined to a submanifold 
of much smaller dimensionality than that of its [[phase space]].

Three features are often referred to as characterizing integrable systems:<ref>{{cite book |first1=N.J. |last1=Hitchin |first2=G.B. |last2=Segal |first3=R.S. |last3=Ward |title=Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces |url=https://books.google.com/books?id=eQ8oAAAAQBAJ |date=2013 |publisher=Oxford University Press |isbn=978-0-19-967677-4 |orig-year=1999}}</ref>
* the existence of a ''maximal'' set of conserved quantities (the usual defining property of '''complete integrability''')
* the existence of '''algebraic''' invariants, having a basis in [[algebraic geometry]] (a property known sometimes as '''algebraic integrability''')
* the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as '''solvability''')

Integrable systems may be seen as very different in qualitative character from more ''generic'' dynamical systems,
which are more typically [[chaos theory|chaotic systems]].  The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over a sufficiently large time.

Many systems studied in physics are completely integrable, in particular, in the [[Hamiltonian system|Hamiltonian]] sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center (e.g., the sun) or two.  Other elementary examples include the motion of a rigid body about its center of mass (the [[Euler top]]) and the motion of an axially symmetric rigid body about a point in its  axis of symmetry (the [[Lagrange top]]).

In the late 1960s, it was realized that there are ''completely integrable'' systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves ([[Korteweg–de Vries equation]]), the [[Kerr effect]] in optical fibres, described by the [[nonlinear Schrödinger equation]], and certain integrable many-body systems, such as the [[Toda lattice]]. The modern theory of integrable systems was revived with the numerical discovery of [[solitons]] by [[Martin Kruskal]] and [[Norman Zabusky]] in 1965, which led to the [[inverse scattering transform]] method in 1967. 

In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the '''leaves''' of the [[Lagrangian foliation]]), and if the flows are complete and the energy level set is compact, this implies the [[Liouville–Arnold theorem]]; i.e.,  the existence of [[action-angle variables]]. General dynamical systems have no such conserved quantities; in the case of autonomous [[Hamiltonian system|Hamiltonian]] systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic.

A key ingredient in characterizing integrable systems is the [[Frobenius theorem (differential topology)|Frobenius theorem]], which states that a system is '''Frobenius integrable''' (i.e., is generated by an integrable distribution) if, locally, it has a [[foliation]] by maximal integral manifolds. But integrability, in the sense of [[dynamical systems]], is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds.

Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set of [[special functions]]; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.

==General dynamical systems==
In the context of differentiable [[dynamical systems]], the notion of '''integrability''' refers to the existence of  invariant, regular [[foliation]]s;  i.e., ones whose leaves are [[embedded submanifold]]s of the smallest possible dimension that are invariant under the  [[Flow (mathematics)|flow]]. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case of [[Hamiltonian mechanics|Hamiltonian systems]], known as '''complete integrability in the sense of [[Liouville]]''' (see below), which is what is most frequently referred to in this context.

An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of [[differential equations]] or [[finite difference| finite difference equations]].

The distinction between integrable and nonintegrable dynamical systems has the qualitative implication of regular motion vs. [[chaotic motion]] and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in an exact form.

==Hamiltonian systems and Liouville integrability==
In the special setting of  [[Hamilton's equations|Hamiltonian systems]], we have the notion of integrability in the [[Joseph Liouville|Liouville]] sense. (See the [[Liouville–Arnold theorem]].) '''Liouville integrability''' means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated with the invariants of the foliation span the tangent distribution.  Another way to state this is that there exists a maximal set of functionally independent [[Poisson bracket |Poisson commuting]] invariants (i.e., independent functions on the phase space whose [[Poisson bracket]]s with the Hamiltonian of the system, and with each other, vanish).

In finite dimensions, if the [[phase space]] is [[Symplectic geometry|symplectic]] (i.e., the center of the Poisson algebra consists only of constants), it must have even dimension <math>2n, </math> and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is <math>n </math>. The leaves of the foliation are [[Lagrangian submanifold|totally isotropic]] with respect to the symplectic form and such a maximal isotropic foliation is called [[Lagrangian submanifold|Lagrangian]]. All ''autonomous'' Hamiltonian systems (i.e. those for which the Hamiltonian and Poisson brackets are not explicitly time-dependent) have at least one invariant; namely, the Hamiltonian itself, whose value along the flow is the energy. If the energy level sets are compact, the leaves of the Lagrangian foliation are [[ torus |tori]], and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical <math> 1 </math>-form are called the action variables, and the resulting canonical coordinates are called [[action-angle variables]] (see below).

There is also a distinction between '''complete integrability''', in the [[Liouville]] sense,  and partial integrability, as well as a notion of [[Superintegrable Hamiltonian system|superintegrability]] and maximal superintegrability.  Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent Poisson commuting invariants is less than maximal (but, in the case of autonomous systems, more than one), we say the system is partially integrable. When there exist further functionally independent invariants, beyond the maximal number that can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is [[Superintegrable Hamiltonian system|superintegrable]]. If there is a regular foliation with one-dimensional leaves (curves), this is called maximally superintegrable.

==Action-angle variables==
When a finite-dimensional Hamiltonian system is completely integrable in the Liouville sense,
and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are [[torus|tori]]. There then exist, as mentioned above, special sets of [[canonical coordinates]] on the [[phase space]]  known as [[action-angle variables]],
such that the invariant tori are the joint level sets of the [[action (physics)|action]]  variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the tori. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.

== The Hamilton&ndash;Jacobi approach ==
In [[canonical transformation]] theory, there is the [[Hamilton&ndash;Jacobi equations|Hamilton&ndash;Jacobi method]], in which solutions to Hamilton's equations are sought by first finding a complete solution of the associated [[Hamilton&ndash;Jacobi equation]]. In classical terminology, this is described as determining a transformation to a canonical set of coordinates consisting of completely ignorable variables; i.e., those in which there is no dependence of the Hamiltonian on a complete set of canonical "position" coordinates, and hence the corresponding canonically conjugate momenta are all conserved quantities.  In the case of compact energy level sets, this is the first step towards determining the [[action-angle variables]]. In the general theory of partial differential equations of [[Hamilton&ndash;Jacobi equations|Hamilton&ndash;Jacobi]] type, a complete solution (i.e. one that depends on ''n'' independent constants of integration, where ''n'' is the dimension of the configuration space), exists in very general cases, but only in the local sense. Therefore, the existence of a complete solution of the [[Hamilton&ndash;Jacobi equation]] is by no means a characterization of complete integrability in the Liouville sense. Most cases that can be "explicitly integrated" involve a complete [[separation of variables]], in which the separation constants provide the complete set of integration constants that are required.  Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.

==Solitons and inverse spectral methods==
A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that [[soliton]]s, which are strongly stable, localized solutions of partial differential equations like the [[Korteweg–de Vries equation]] (which describes 1-dimensional non-dissipative fluid dynamics in shallow basins), could be understood by viewing these equations as infinite-dimensional integrable Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, the  [[inverse scattering transform]] and more general inverse spectral methods (often reducible to [[Riemann–Hilbert problem]]s),
which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations.

The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" (in a suitably generalized sense) is invariant under the evolution, cf. [[Lax pair]].  This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable.  In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to '''completely ignorable coordinates''', in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.

== Hirota bilinear equations and  τ-functions==

Another viewpoint that arose in the modern theory of integrable systems originated in
a calculational approach pioneered by ''Ryogo Hirota'',<ref>{{cite journal |first=R. |last=Hirota |title=Reduction of soliton equations in bilinear form |journal=Physica D: Nonlinear Phenomena |volume=18 |issue=1–3 |pages=161–170 |year=1986 |doi=10.1016/0167-2789(86)90173-9 |bibcode=1986PhyD...18..161H}}</ref> which involved replacing
the original nonlinear dynamical system with a bilinear system of constant coefficient
equations for an auxiliary quantity, which later came to be known as the 
[[Tau function (integrable systems)|τ-function]]. These are now referred to as the ''Hirota equations''. Although originally appearing just as a calculational device, without any clear relation
to the [[inverse scattering transform|inverse scattering]] approach, or the Hamiltonian structure, this nevertheless gave a very direct method from which important classes of solutions such as [[solitons]] could be derived.

Subsequently, this was interpreted by [[Mikio Sato]]<ref name="Sato">{{cite journal |first=M. |last=Sato |title=Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds |journal=Kokyuroku, RIMS, Kyoto University |volume=439 |pages=30–46 |year=1981 |hdl=2433/102800 |url=https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/102800/1/0439-5.pdf}}</ref> and his students,<ref name="DJKM1">{{cite journal |first1=E. |last1=Date |first2=M. |last2=Jimbo |first3=M. |last3=Kashiwara |first4=T. |last4=Miwa |title=Operator approach to the Kadomtsev-Petviashvili equation III |journal=Journal of the Physical Society of Japan |volume=50 |issue=11 |pages=3806–12 |year=1981 |doi=10.1143/JPSJ.50.3806}}</ref><ref name="DJKM2">{{cite journal |first1=M. |last1=Jimbo |first2=T. |last2=Miwa |title=Solitons and infinite-dimensional Lie algebras |journal=Publ. Res. Inst. Math. Sci. |volume=19 |issue=3 |pages=943–1001 |year=1983 |doi=10.2977/prims/1195182017 |url=https://www.jstage.jst.go.jp/article/kyotoms1969/19/3/19_3_943/_article|doi-access=free }}</ref> at first for the case of 
integrable hierarchies of PDEs, such as the [[Kadomtsev–Petviashvili equation|Kadomtsev–Petviashvili]] hierarchy, but then 
for much more general classes of integrable hierarchies, as a sort of ''universal phase space'' approach, in which, typically, the commuting dynamics were viewed simply as determined by a fixed (finite or infinite) abelian [[group action]] on a (finite or infinite) [[Grassmannian|Grassmann manifold]].  
The τ-function was viewed as the [[determinant]]
of a [[projection operator]] from elements of the [[group orbit]] to some ''origin'' within the Grassmannian,
and the ''Hirota equations'' as expressing the [[Plücker embedding|Plücker relations]], characterizing the 
Plücker embedding of the Grassmannian in the projectivization of a suitably 
defined (infinite) [[exterior algebra|exterior space]], viewed as a [[Fock space|fermionic Fock space]].

==Quantum integrable systems==
There is also a notion of quantum integrable systems.

In the quantum setting, functions on phase space must be replaced by [[self-adjoint operators]] on a [[Hilbert space]], and the notion of Poisson commuting functions replaced by commuting operators. The notion of conservation laws must be specialized to [[Principle of locality|local]] conservation laws.<ref>{{cite journal | last1=Calabrese | first1=Pasquale | last2=Essler | first2=Fabian H L | last3=Mussardo | first3=Giuseppe | title=Introduction to 'Quantum Integrability in Out of Equilibrium Systems' | journal=Journal of Statistical Mechanics: Theory and Experiment | publisher=IOP Publishing | volume=2016 | issue=6 | date=2016-06-27 | issn=1742-5468 | doi=10.1088/1742-5468/2016/06/064001 | page=064001| bibcode=2016JSMTE..06.4001C | s2cid=124170507 | url=https://ora.ox.ac.uk/objects/uuid:8bf032d9-9d7a-4658-8238-6f297b2a0598 }}</ref> Every [[Hamiltonian (quantum mechanics)|Hamiltonian]] has an infinite set of conserved quantities given by projectors to its energy [[eigenstates]]. However, this does not imply any special dynamical structure.

To explain quantum integrability, it is helpful to consider the free particle setting.  Here all dynamics are one-body reducible.  A quantum system is said to be integrable if the dynamics are two-body reducible. The [[Yang–Baxter equation]] is a consequence of this reducibility and leads to trace identities which provide an infinite set of conserved quantities. All of these ideas are incorporated into the [[quantum inverse scattering method]] where the algebraic [[Bethe ansatz]] can be used to obtain explicit solutions. Examples of quantum integrable models are the [[Lieb–Liniger model]], the [[Hubbard model]] and several variations on the [[Heisenberg model (quantum)|Heisenberg model]].<ref>{{cite book | author-link=Vladimir Korepin |first1=V.E. |last1=Korepin |first2=N.M. |last2=Bogoliubov |first3=A.G. |last3=Izergin  | title=Quantum Inverse Scattering Method and Correlation Functions| publisher=Cambridge University Press | year = 1997 | isbn=978-0-521-58646-7}}</ref> Some other types of quantum integrability are known in explicitly time-dependent quantum problems, such as the driven Tavis-Cummings model.<ref name='sinitsyn-16pra1'>{{cite journal|doi=10.1103/PhysRevA.93.063859|title=Solvable multistate model of Landau-Zener transitions in cavity QED |first1=N.A. |last1=Sinitsyn |first2=F. |last2=Li |journal=[[Phys. Rev. A]]|volume=93|issue=6|year=2016|pages= 063859|bibcode=2016PhRvA..93f3859S|arxiv=1602.03136|s2cid=119331736}}</ref>

==Exactly solvable models==
In physics, completely integrable systems, especially in the infinite-dimensional setting, are often referred to as exactly solvable models.  This obscures the distinction between integrability, in the Hamiltonian sense, and the more general dynamical systems sense.

There are also exactly solvable models in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the [[Bethe ansatz]] approach, in its modern sense, based on the [[Yang–Baxter equation]]s and the [[quantum inverse scattering method]], provide quantum analogs of the inverse spectral methods. These are equally important in the study of solvable models in statistical mechanics.

An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have some "known" functions available, in terms of which the solutions may be expressed.  This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems. {{citation needed|date=May 2015}}

==List of some well-known integrable systems==
;Classical mechanical systems
* Calogero–Moser–Sutherland model<ref>{{cite journal |author-link=Francesco Calogero |first=F. |last=Calogero |year=2008 |title=Calogero-Moser system |journal=Scholarpedia |volume=3 |issue=8 |page=7216|doi=10.4249/scholarpedia.7216 |bibcode=2008SchpJ...3.7216C |doi-access=free }}</ref>
* [[Central force]] motion ([[exact solutions of classical central-force problems]])
* [[Anosov flow|Geodesic motion on ellipsoids]]
* [[Harmonic oscillator]]
* Integrable Clebsch and Steklov systems in fluids
* [[Lagrange, Euler, and Kovalevskaya tops]]
* [[Carl Neumann|Neumann oscillator]]
* Two center [[Newtonian gravitation]]al motion
;Integrable lattice models
* Ablowitz–Ladik lattice
* [[Toda lattice]]
* [[Volterra lattice]]
;Integrable systems in 1 + 1 dimensions
* [[AKNS system]]
* [[Benjamin–Ono equation]]
* [[Boussinesq equation (water waves)]]
* [[Camassa–Holm equation]]
* [[Classical Heisenberg ferromagnet model (spin chain)]]
* [[Degasperis–Procesi equation]]
* [[Dym equation]]
* [[Garnier integrable system]]
* [[Kaup–Kupershmidt equation]]
* Krichever–Novikov equation
* [[Korteweg–de Vries equation]]
* [[Landau–Lifshitz equation (continuous spin field)]]
* [[Nonlinear Schrödinger equation]]
* [[Nonlinear sigma models]]
* [[Sine–Gordon equation]]
* [[Thirring model]]
* [[Three-wave equation]]
;Integrable PDEs in 2 + 1 dimensions
* [[Davey–Stewartson equation]]
* [[Ishimori equation]]
* [[Kadomtsev–Petviashvili equation]]
* [[Novikov–Veselov equation]]
;Integrable PDEs in 3 + 1 dimensions
* The [[Belinski–Zakharov transform]] generates a Lax pair for the [[Einstein field equation]]s; general solutions are termed [[gravitational soliton]]s, of which the [[Schwarzschild metric]], the [[Kerr metric]] and some [[gravitational wave]] solutions are examples.
;Exactly solvable statistical lattice models
* [[8-vertex model]]
* [[Gaudin model]]
* [[Ising model]] in 1- and 2-dimensions
* [[Ice-type model]] of Lieb
* [[Quantum Heisenberg model]]

==See also==
* [[Hitchin system]]
* [[Pentagram map]]

===Related areas===
* [[Mathematical physics]]
* [[Soliton]]
* [[Painleve transcendents]]
* [[Statistical mechanics]]
* [[Integrable algorithm]]

=== Some key contributors (since 1965) ===
{{columns-list|colwidth=20em|
* [[Mark Ablowitz]]
* [[Rodney Baxter]]
* [[Percy Deift]]
* Leonid Dickey
* [[Vladimir Drinfeld]]
* [[Boris Dubrovin (mathematician)|Boris Dubrovin]]
* [[Ludvig Faddeev]]
* [[Hermann Flaschka]]
* [[Israel Gel'fand]]
* [[Alexander Its]]
* [[Michio Jimbo]]
* [[Igor Krichever |Igor M. Krichever]]
* [[Martin Kruskal]]
* [[Peter Lax]]
* Vladimir Matveev
* Henry McKean
* [[Robert Miura]]
* [[Tetsuji Miwa]]
* [[Alan C. Newell|Alan Newell]]
* [[Nicolai Reshetikhin]]
* Aleksei Shabat
* [[Evgeny Sklyanin]]
* [[Mikio Sato]]
* [[Elliott H. Lieb]]
* [[Graeme Segal]]
* George Wilson
* [[Vladimir E. Zakharov]]
}}


==References==
*{{cite book |author-link=Vladimir Arnold |first=V.I. |last=Arnold | title=Mathematical Methods of Classical Mechanics |edition=2nd | publisher=Springer | year=1997 | isbn=978-0-387-96890-2 | url-access=registration | url=https://archive.org/details/mathematicalmeth0000arno }}
*{{cite book |author-link=Michèle Audin |first=M. |last=Audin | title = Spinning Tops: A Course on Integrable Systems |
publisher= [[Cambridge University Press]] | series = Cambridge Studies in Advanced Mathematics 
| volume = 51 | year = 1996 | isbn = 978-0521779197}}
*{{cite book |first1=O. |last1=Babelon |first2=D. |last2=Bernard |first3=M. |last3=Talon | title=Introduction to classical integrable systems | publisher=[[Cambridge University Press]] | year = 2003 | isbn=0-521-82267-X |doi=10.1017/CBO9780511535024}}
*{{cite book |author-link=Rodney Baxter |first=R.J. |last=Baxter |title=Exactly solved models in statistical mechanics | publisher=Academic Press | year = 1982 | isbn=978-0-12-083180-7}}
*{{cite book |first=M. |last=Dunajski | title=Solitons, Instantons and Twistors| publisher=[[Oxford University Press]]| year = 2009|isbn=978-0-19-857063-9 }}
*{{cite book |author-link=Ludvig Faddeev |first1=L.D. |last1=Faddeev |author2-link=Leon Takhtajan |first2=L.A. |last2=Takhtajan | title =Hamiltonian Methods in the Theory of Solitons | publisher= Addison-Wesley | year=1987 |isbn=978-0-387-15579-1 }}
*{{cite book |author-link=Anatoly Fomenko |first=A.T. |last=Fomenko |title=Symplectic Geometry. Methods and Applications |publisher=Gordon and Breach |year=1995 |isbn=978-2-88124-901-3 |edition=2nd }}
*{{cite book|author-link=Anatoly Fomenko |first1=A.T. |last1=Fomenko |first2=A.V. |last2=Bolsinov |title=Integrable Hamiltonian Systems: Geometry, Topology, Classification |publisher=Taylor and Francis |year=2003 |isbn=978-0-415-29805-6 }}
*{{cite book |author-link=Herbert Goldstein |first=H. |last=Goldstein | title =Classical Mechanics |edition=2nd | publisher= Addison-Wesley | year=1980 |isbn= 0-201-02918-9}}
*{{cite book |editor-link=John Harnad |editor-first=J. |editor-last=Harnad |editor2-link=Pavel Winternitz |editor2-first=P. |editor2-last=Winternitz |editor3-link=Gert Sabidussi |editor3-first=G. |editor3-last=Sabidussi | title=Integrable Systems: From Classical to Quantum| publisher=[[American Mathematical Society]] | year = 2000 | isbn=0-8218-2093-1 }}
* {{cite book |author-link=John Harnad |first1=J. |last1=Harnad |first2=F. |last2=Balogh | title=Tau functions and Their Applications | publisher=[[Cambridge University Press]] | year = 2021 | series = Cambridge Monographs on Mathematical Physics |
isbn=9781108492683 |doi= 10.1017/9781108610902|s2cid=222379146 }}
*{{cite book |first1=J. |last1=Hietarinta |author2-link=Nalini Joshi |first2=N. |last2=Joshi |first3=F. |last3=Nijhoff | title=Discrete systems and integrability | publisher=[[Cambridge University Press]] | year = 2016 | isbn=978-1-107-04272-8 |doi=10.1017/CBO9781107337411|bibcode=2016dsi..book.....H }}
*{{cite book |author-link=Vladimir Korepin |first1=V. E. |last1=Korepin |first2=N.M. |last2=Bogoliubov |first3=A.G. |last3=Izergin | title=Quantum Inverse Scattering Method and Correlation Functions| publisher=[[Cambridge University Press]] | year = 1997 | isbn=978-0-521-58646-7}}
*{{cite book | author-link=Valentin Afraimovich |first1=V.S. |last1=Afrajmovich |author2-link=Vladimir Arnold |first2=V.I. |last2=Arnold |author3-link=Yulij Ilyashenko |first3=Yu. S. |last3=Il'yashenko |first4=L.P. |last4=Shil'nikov | title=Dynamical Systems V | publisher=Springer | isbn=3-540-18173-3 }}
*{{cite book |first=Giuseppe |last=Mussardo | title=Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics | year=2010 | publisher=[[Oxford University Press]] | isbn=978-0-19-954758-6}}
*{{cite book |author-link=Gennadi Sardanashvily |first=G. |last=Sardanashvily | title=Handbook of Integrable Hamiltonian Systems | publisher=URSS | year=2015 | isbn=978-5-396-00687-4}}

== Further reading ==
*{{cite web |author-link=Alexander Beilinson |first1=A. |last1=Beilinson |author2-link=Vladimir Drinfeld |first2=V. |last2=Drinfeld |title=Quantization of Hitchin's integrable system and Hecke eigensheaves |url=http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf}}<!-- the article needs to discuss the connection to algebraic geometry. -- Taku -->
*{{cite magazine |author-link=Ron Donagi |last1=Donagi |first1=R. |last2=Markman|first2=E.|year=1996|title=Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles.|magazine=Integrable systems and quantum groups|series=Lecture Notes in Mathematics |volume=1620 |publisher=Springer|pages=1–119|doi=10.1007/BFb0094792|isbn=978-3-540-60542-3 }}
*{{cite journal |first1=Kiran G. |last1=Sonnad |first2=John R. |last2=Cary |year=2004 |title=Finding a nonlinear lattice with improved integrability using Lie transform perturbation theory |journal=Physical Review E |volume=69 |issue=5 |page=056501 |doi=10.1103/PhysRevE.69.056501|pmid=15244955 |bibcode=2004PhRvE..69e6501S }}

==External links==
{{Commons category|Integrable systems}}

* {{springerEOM|title=Integrable system|id=Integrable_system&oldid=43547}}
* [http://www.side-conferences.net/ "SIDE - Symmetries and Integrability of Difference Equations"], a conference devoted to the study of integrable difference equations and related topics.<ref>{{cite book |first1=Peter A. |last1=Clarkson |first2=Frank W. |last2=Nijhoff |title=Symmetries and Integrability of Difference Equations |url=https://books.google.com/books?id=oi_jvhD1YU0C |date=1999 |publisher=Cambridge University Press |isbn=978-0-521-59699-2 |series=London Mathematical Society |volume=255}}</ref>

==Notes==
{{reflist}}

{{Integrable systems}}

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[[Category:Hamiltonian mechanics]]
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