Symmetry Analysis, Integrability and Completeness of Geodesics of a Double of the Affine Lie Group of the Real Line

Abstract

We investigate Lie point symmetries of a system of four nonlinear secondorder ordinary differential equations (ODEs), appropriated to the geodesics of a Drinfel'd double Lie group of the affine Lie group of $\mathbb{R}$. The first integrals associated with Lie point symmetries are obtained by utilizing the constructive method due to Wafo Soh and Mahomed [16]. This method deals with integrability when the symmetry vector fields and the operator associated to the system are unconnected. In certain cases we obtain the explicit expressions of the geodesics. We also show that the geodesics are not complete.