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Remarks on integrable systems. (English) Zbl 1382.34002
Summary: The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an \(n\)-dimensional space, which admit the algebra of symmetry fields of dimension \(\leqslant n\). Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.

MSC:
34A05 Explicit solutions, first integrals of ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
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