Compact objects with scalar charge embedded in a magnetic or electric field in Einstein-Maxwell-dilaton theories

In this paper, we generalize the Schwarzschild-Melvin solution within Einstein-Maxwell-dilaton theories to include non-null scalar charges, while remaining embedded in a magnetic or electric field $\grave{\mathit{\text{a}}}\mathit{\text{la}}\mathit{\text{Melvin}}$. We then use this general solution to obtain the solution in the specific case of Entangled Relativity after a conformal transformation. This notably enables us to verify that the analytical solution used in [Arruga & Minazzoli 2021] in order to represent compact objects such as neutron stars in Entangled Relativity is indeed a good approximation of the exact solutions of Entangled Relativity when the background field goes to zero.