  We show that in collisions with neutron rich heavy ions at energies around
the production threshold K^0 and K^+ yields might probe the isospin dependent
part of the nuclearEquation of State (EoS) at high baryon densities. In
particular we suggest the K^0/K^+ ratio as a promising observable. Results
obtained in a fully covariant relativistic transport approach are presented for
central Au+Au collisions in the beam energy range 0.8-1.8~AGeV. The focus is
put on the EoS influence which goes beyond the "collision-cascade" picture. The
isovector part of the in-medium interaction affects the kaon multiplicities via
two mechanisms: i) a "symmetry potential" effect, i.e. a larger neutron
repulsion in n-rich systems (isospin fractionation); ii) a "threshold" effect,
due to the change in the self-energies of the particles involved in inelastic
processes. Genuine relativistic contributions are revealed, that could allow to
directly ``measure'' the Lorentz structure of the effective isovector
interaction.
