The family of regular interiors for non-rotating black holes with $T^0_0 = T^1_1$

We find the general solution for the spacetimes describing the interior of static black holes with an equation of state of the type $ T^0_0 = T^1_1 $ ($T$ being the stress-energy tensor). This form is the one expected from taking into account different quantum effects associated with strong gravitational fields. We recover all the particular examples found in the literature. We remark that all the solutions found follow the natural scheme of an interior core linked smoothly with the exterior solution by a transient region. We also discuss their local energy properties and give the main ideas involved in a possible generalization of the scheme, in order to include other realistic types of sources.