First-order perturbations of Gödel-type metrics in non-dynamical Chern–Simons modified gravity

G\"{o}del-type metrics that are homogeneous in both space and time remain, like the Schwarzschild metric, consistent within Chern-Simons modified gravity; this is true non-dynamical dynamical frameworks, each of which involves an additional pseudoscalar field coupled to Pontryagin density. In paper, we consider stationary first-order perturbations these framework. Under certain assumptions find analytical solutions perturbed equations. The equations break translational cylindrical symmetries unperturbed metrics. effective potential controlling planar geodesic orbits also affected by perturbation parameter, changes equilibrium radii for massive particles massless photons.