Lie point symmetries of the geodesic equations of the Godel's metric

Lie point symmetries of the geodesic equations of the Godel's metric are found. These form a ten dimensional Lie algebra. The Lie algebra contains a maximal seven-dimensional solvable sub- algebra. It also contains five dimensional subalgebra of isometries of the metric. The isometries are used to reduce the order of the geodesic system by one. The time-like trajectories of the Godel's metric are then derived and their graphs in the (r, phi) plane are displayed showing some interesting features of the dynamics in this universe.