We consider the geodesic system for Gödel’s metric as a toy model and solve it analytically using its Lie point symmetries. It is shown that the differential invariants of these symmetries reduce the second-order non-linear system to a single second-order ordinary differential equation (ODE). Invariance of the latter under a one-dimensional Lie point symmetry group reduces it to an integrable first-order ODE. A complete solution of the system is then achieved. The sub-algebras of Noether symmetries and isometries are then found with their corresponding first integrals.