  Results on $K^+$ and $K^0$ meson production in $^{96}_{44}$Ru +
$^{96}_{44}$Ru and $^{96}_{40}$Zr + $^{96}_{40}$Zr collisions at a beam kinetic
energy of 1.528$A$ GeV, measured with the FOPI detector at GSI-Darmstadt, are
investigated as a possible probe of isospin effects in high density nuclear
matter. The measured double ratio ($K^+/K^0$)$_{Ru}$/($K^+/K^0$)$_{Zr}$ is
compared to the predictions of a thermal model and a Relativistic Mean Field
transport model using two different collision scenarios and under different
assumptions on the stiffness of the symmetry energy. We find a good agreement
with the thermal model prediction and the assumption of a soft symmetry energy
for infinite nuclear matter while more realistic transport simulations of the
collisions show a similar agreement with the data but also exhibit a reduced
sensitivity to the symmetry term.
