  In the version \cite{Fiziev14} of this paper we presented for the first time
the basic equations and relations for relativistic static spherically symmetric
stars (SSSS) in the model of minimal dilatonic gravity (MDG). This model is
{\em locally} equivalent to the f(R) theory of gravity and gives an alternative
description of the effects of dark matter and dark energy using the
Branse-Dicke dilaton $\Phi$. To outline the basic properties of the MDG model
of SSSS and to compare them with general relativistic results, in the present
paper we use the relativistic equation of state (EOS) of neutron matter as an
ideal Fermi neutron gas at zero temperature. We overcome the well-known
difficulties of the physics of SSSS in the f(R) theories of gravity
\cite{Felice10,Berti} applying novel highly nontrivial nonlinear boundary
conditions, which depend on the global properties of the solution and on the
EOS. We also introduce two pairs of new notions: cosmological-energy-pressure
densities and dilaton-energy-pressure densities, as well as two new EOS for
them: cosmological EOS (CEOS) and dilaton EOS (DEOS). Special attention is paid
to the dilatonic sphere (in brief -- disphere) of SSSS, introduced in this
paper for the first time. Using several realistic EOS for neutron star (NS):
SLy, BSk19, BSk20 and BSk21, and current observational two-solar-masses-limit,
we derive an estimate for scalar-field-mass $m_\Phi \sim 10^{-13} eV/c^2 \div
4\times 10^{-11} eV/c^2$. Thus, the present version of the paper reflects some
of the recent developments of the topic.
