  We present a resum\'e on the modified theory of gravity, called
pseudo-complex General Relativity (pc-GR). It is the second in a series of
papers, where the first one (Boller et al. 2019, referred to as paper I)
discussed the observational consequences of pc-GR. In this paper, we
concentrate on the underlying theory. PC-GR involves an algebraic extension of
the standard theory of GR and it depends on two phenomenological parameters. An
element included in pc-GR that is not present in standard GR is the
energy-momentum tensor corresponding to an anisotropic ideal fluid, which we
call dark energy. The two parameters are related to the coupling of mass to the
dark energy and its fall-off as a function of r. The consequences and
predictions of this theory will be discussed in the context of the
observational results of the Even Horizon Telescope, expected soon. Our main
result is that due to the accumulation of dark energy near a large mass, the
modified theory predicts a dark ring followed by a bright ring in the emission
profile of the accretion disc. We also discuss the light ring in the equatorial
plane.
