This work investigates the geodesic structure of a Finslerian extension of Kerr space-time constructed within the Barthel–Randers framework, focusing on the effects of small direction-dependent deviations from general relativity. A dimensionless deformation parameter ξ ˜ = ξ / M 2 is introduced to model leading-order space-time anisotropy while preserving axial symmetry and asymptotic flatness. Using the osculating Riemannian approach, perturbative analytical expressions for null and time-like geodesics are derived, yielding explicit corrections to the photon-sphere and innermost stable circular orbit (ISCO) radii in terms of the spin parameter a ˜ and ξ ˜ . A leading-order spin–anisotropy degeneracy is identified, in the sense that correlated variations of a ˜ and ξ ˜ generate comparable shifts in the photon sphere and ISCO radii. For small anisotropic deformations | ξ ˜ | ≲ 0.1 , these corrections are comparable to those induced by variations of the Kerr spin parameter, indicating a degeneracy at the level of orbital structure. A direct connection to observable quantities such as black-hole shadows or X-ray spectra requires a full analysis of the photon region and radiation transfer, which is beyond the scope of the present work. The results suggest that Finsler-type space-time anisotropy can produce systematic modifications of strong-field dynamics and motivate further investigation of anisotropic effects in rotating black-hole geometries.
Z. Nekouee (Mon,) studied this question.