Nonlinear Dynamics

, Volume 74, Issue 4, pp 969–989 | Cite as

Symmetries of second-order systems of ODEs and integrability

  • Muhammad Ayub
  • F. M. Mahomed
  • Masood Khan
  • M. N. Qureshi
Original Paper


We present a method for finding a complete set of kth-order (k≥2) differential invariants including bases of invariants corresponding to vector fields in three variables of four-dimensional real Lie algebras. As a consequence, we provide a complete list of second-order differential invariants and canonical forms for vector fields of four-dimensional Lie algebras and their admitted regular systems of two second-order ODEs. Moreover, we classify invariant representations of these canonical forms of ODEs into linear, partial linear, uncoupled, and partial uncoupled cases. In addition, we give an integration procedure for invariant representations of canonical forms for regular systems of two second-order ODEs admitting four-dimensional Lie algebras.


Systems of ODEs Canonical forms Lie symmetries Invariant representation Integrability 



FMM thanks the National Research Foundation of South Africa for an enabling research grant. We are grateful to the referees for useful comments.


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Muhammad Ayub
    • 1
  • F. M. Mahomed
    • 2
  • Masood Khan
    • 3
  • M. N. Qureshi
    • 4
  1. 1.Department of MathematicsCOMSATS Institute of Information TechnologyAbbottabadPakistan
  2. 2.Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied MathematicsUniversity of the WitwatersrandWitsSouth Africa
  3. 3.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  4. 4.Department of MathematicsAzad Jammu and Kashmir UniversityMuzaffarabadPakistan

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