  Subthreshold kaon (K$^{0}$ and K$^{+}$) production in neutron-rich nuclear
reactions to probe the kaon-nucleon interaction in nuclear medium and to
extract the isospin dependent part of the nuclear equation of state at
high-baryon densities, is investigated within an isospin and momentum dependent
transport model. A repulsive kaon-nucleon potential is implemented in the model
through fitting the flow data and inclusive spectra in heavy-ion collisions,
which enhances the energetic kaon emission squeezed out in the reaction zone
and leads to a variation of the high-momentum spectrum of the K$^{0}$/K$^{+}$
yields. It is found that the stiffness of nuclear symmetry energy plays a
significant role on the isospin ratio with decreasing the incident energy and a
hard symmetry energy has a larger value of the K$^{0}$/K$^{+}$ ratio in the
domain of subthreshold energies.
