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Journal of Applied Mathematics
Volume 2013, Article ID 147921, 15 pages
Research Article

Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

1Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan
2Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa

Received 27 November 2012; Accepted 20 January 2013

Academic Editor: Mehmet Pakdemirli

Copyright © 2013 Muhammad Ayub et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present a systematic procedure for the determination of a complete set of kth-order ( ) 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 kth-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 . 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 kth-order ( ) 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.