Analytic treatment of complete geodesics in a static cylindrically symmetric conformal spacetime

We consider the motion of test particles and light rays in a static cylindrically symmetric conformal spacetime given by Said et al [1]. We derive the equations of motion and present their analytical solutions in terms of the Weierstrass {\wp} function and the Kleinian {\sigma} function. Using parametric diagrams and effective potentials we analyze the possible orbits and characterize them in terms of the energy and the angular momentum of the test particles. Finally we show some examples of orbits.