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A vaguely defined yet very popular notion which may mean one of the following:

for distributions);

  • For differential equations (both ordinary and partial) and dynamical systems:
  1. a possibility to find solution in a given class of functions (Darbouxian integrability, Liouville integrability, integrability in quadratures etc.) or just in some closed form;
  2. existence of one or more first integrals, functions which remain constant along solutions;
  3. preservation of some additional structures (e.g., Hamiltonian systems are sometimes called integrable to distinguish them from dissipative systems);
  4. complete integrability for Hamiltonian systems means existence of the maximal possible number of first integrals in involution.

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Integrability. Encyclopedia of Mathematics. URL: