Integrability

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

• For functions - existence of the integral in some sense (Riemann integrability, Lebesgue integrability, improper integrals etc.);
• For geometric structures and partial differential equations - conditions guaranteeing existence of solutions (Frobenius integrability condition

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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