Symmetries of geodesic motion in Gödel-type spacetimes

In this paper, we study Noether gauge symmetries of geodesic motion for geodesic Lagrangian of four classes of metrics of Gödel-type spacetimes for which we calculated the Noether gauge symmetries for all classes I-IV, and find the first integrals of corresponding classes to derive a complete characterization of the geodesic motion. Using the obtained expressions for $\dot{t}, \dot{r}, \dotϕ$ and $\dot{z}$ of each classes I-IV which depends essentially on two independent parameters $m$ and $w$, we explicitly integrated the geodesic equations of motion for the corresponding Gödel-type spacetimes.

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