  We employ the extended Nambu-Jona-Lasinio, linear-$\sigma$ models, and the
density-dependent model with chiral limits to work out the mean fields and
relevant properties of nuclear matter. To have the constraint from the data, we
reexamine the Dirac optical potentials and symmetry potential based on the
relativistic impulse approximation (RIA). Unlike the extended NJL and the
density-dependent models with the chiral limit in terms of the vanishing scalar
density, the extended linear-$\sigma$ model with a sluggish changing scalar
field loses the chiral limit at the high density end. The various scalar fields
can characterize the different Schr\"odinger-equivalent potentials and kinetic
symmetry energy in the whole density region and the symmetry potential in the
intermediate density region. The drop of the scalar field due to the chiral
restoration results in a clear rise of the kinetic symmetry energy. The chiral
limit in the models gives rise to the softening of the symmetry potential and
thereof the symmetry energy at high densities.
