  Neutron stars with the isovector scalar $\delta$-field are studied in the
framework of the relativistic mean field ($RMF$) approach in a pure nucleon
plus lepton scheme. The $\delta$-field leads to a larger repulsion in dense
neutron-rich matter and to a definite splitting of proton and neutron effective
masses. Both features are influencing the stability conditions of the neutron
stars. Two parametrizations for the effective nonlinear Lagrangian density are
used to calculate the nuclear equation of state ($EOS$) and the neutron star
properties, and compared to correlated Dirac-Brueckner results. We conclude
that in order to reproduce reasonable nuclear structure and neutron star
properties within a $RMF$ approach a density dependence of the coupling
constants is required.
