Original Paper

Nonlinear Dynamics

, Volume 74, Issue 4, pp 969-989

First online:

Symmetries of second-order systems of ODEs and integrability

  • Muhammad AyubAffiliated withDepartment of Mathematics, COMSATS Institute of Information Technology Email author 
  • , F. M. MahomedAffiliated withDifferential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand
  • , Masood KhanAffiliated withDepartment of Mathematics, Quaid-i-Azam University
  • , M. N. QureshiAffiliated withDepartment of Mathematics, Azad Jammu and Kashmir University

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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