## Second-order systems of ODEs admitting three-dimensional Lie algebras and integrability.(English)Zbl 1266.34054

Summary: We present a systematic procedure for the determination of a complete set of $$k$$th-order ($$k \geq 2$$) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two $$k$$th-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case of $$k = 2$$ and 31 classes for the case of $$k \geq 3$$. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of two $$k$$th-order ($$k \geq 3$$) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

### MSC:

 34C14 Symmetries, invariants of ordinary differential equations
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### References:

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