Compact objects with primary hair in shift and parity symmetric beyond Horndeski gravities

In this work, we delve into the model of the shift symmetric and parity-preserving beyond Horndeski theory in all its generality. We present an explicit algorithm to extract static and spherically symmetric black holes with primary scalar charge adhering to the conservation of the Noether current emanating from the shift symmetry. We show that when the functionals G₂ and G₄ of the theory are linearly dependent, analytic homogeneous black-hole solutions exist, which can become regular by virtue of the scalar charge contribution. Such geometries can easily enjoy the preservation of the weak energy conditions, elevating them into healthier compact objects than most hairy black holes in modified theories of gravity. Finally, we revisit the concept of disformal transformations as a solution-generating mechanism and discuss the case of generic G₂ and G₄ functionals.