Symmetries and Integrating Factors

Abstract

Cheb-Terrab and Roche (J. Sym. Comp. 27 (1999), 501–519) presented what they termed a systematic algorithm for the construction of integrating factors for second order ordinary differential equations. They showed that there were instances of ordinary differential equations without Lie point symmetries which were solvable with this algorithm. We demonstrate that the existence of integrating factors is paralleled by the existence of suitable Lie symmetries which enable one to reduce the equations to quadratures thereby emphasising the fact that integrability relies upon symmetry.