  We present a derivation of the basic equations and boundary conditions for
relativistic static spherically symmetric stars (SSSS) in the model of minimal
dilatonic gravity (MDG) which offers an alternative and simultaneous
description of the effects of dark matter (DM) and dark energy (DE) using one
dilaton field $\Phi$. The numerical results for a realistic equation of state
(EOS) MPA1 of neutron matter are presented for the first time. The three very
different scales, the Compton length of the scalar field $\lambda_\Phi$, the
star's radius $r^*$, and the finite radius of the MDG Universe $r_{U}$ are a
source of numerical difficulties. Owing to the introduction of a new dark
scalar field $\varphi=\ln(1+\ln\Phi)$, we have been able to study numerically
an unprecedentedly large interval of $\lambda_\Phi$ and have discovered the
existence of $\lambda_\Phi^{crit}\approx 2.1$\ km for a neutron star with MPA1
EOS. This is related to the bifurcation of the physical domain in the phase
space of the system. Some novel physical consequences are discussed.
