Nonvacuum solutions in Cotton theory

Cotton theory (CT) introduces a higher derivative extension of general relativity (GR) characterized by third-rank field equations. Recently, key distinctions between CT and GR concerning wave and vacuum solutions have been highlighted in Gürses et al. [Wave metrics in the Cotton and conformal Killing gravity theories, Phys. Rev. D 110, 084082 (2024).] and Altas and Tekin [Vanishing of conserved charges in Cotton gravity, Phys. Rev. D 111, L021503 (2025).]. In this study, two particular nonvacuum solutions of CT are investigated within its Codazzi formulation. The motivation is to reveal how this theory might account for or adapt to the effects of nonvacuum sources, and whether it can provide new insights into the behavior of both singular and regular black holes in astrophysical contexts. It is shown that CT generalizes the Kiselev and Dymnikova solutions in GR. Some aspects of the generalized solutions, in particular concerning singularities, thermodynamics, and geodesics, are addressed in comparison to GR.