(Regular) Black holes in conformal Killing gravity coupled to nonlinear electrodynamics and scalar fields

In this work, we explore black hole and regular black hole solutions in the recently proposed Conformal Killing Gravity (CKG). This theory is of third order in the derivatives of the metric tensor and essentially satisfies three theoretical criteria for gravitational theories beyond General Relativity (GR). The criteria essentially stipulate the following, that one should: (i) obtain the cosmological constant as an integration constant; (ii) derive the energy conservation law as a consequence of the field equations, rather than assuming it; (iii) and not necessarily consider conformally flat metrics as vacuum solutions. In fact, existing modified theories of gravity, including GR, do not simultaneously fulfil all of these three criteria. Here, we couple CKG to nonlinear electrodynamics (NLED) and scalar fields, and we explore solutions of black holes and regular black holes. More specifically, by solving the field equations of CKG, we find specific forms for the NLED Lagrangian, the scalar field and the field potential, and analyse the regularity of the solutions through the Kretschmann scalar. We find generalizations of the Schwarschild--Reissner-Nordstr\"{o}m--AdS solutions, and consequently further extend the class of (regular) black hole solutions found in the literature.