Canonical Forms and Their Integrability for Systems of Three 2nd-Order ODEs

Differential invariants and their corresponding canonical forms for systems of three 2nd-order ODEs possessing three-dimensional Lie algebras are constructed. Their extension up to kth-order system of three 2nd-order ODEs is presented. Furthermore singularity in invariant structure for the canonical forms is investigated. In addition integrability of these canonical forms is discussed. Illustrative physical examples from mechanics of system of particles are provided