  The rapidity dependence of the single- and double- neutron to proton ratios
of nucleon emission from isospin-asymmetric but mass-symmetric reactions Zr+Ru
and Ru+Zr at energy range $100 \sim 800$ A MeV and impact parameter range
$0\sim 8$ fm is investigated. The reaction system with isospin-asymmetry and
mass-symmetry has the advantage of simultaneously showing up the dependence on
the symmetry energy and the degree of the isospin equilibrium. We find that the
beam energy- and the impact parameter dependence of the slope parameter of the
double neutron to proton ratio ($F_D$) as function of rapidity are quite
sensitive to the density dependence of symmetry energy, especially at energies
$E_b\sim 400$ A MeV and reduced impact parameters around 0.5. Here the symmetry
energy effect on the $F_D$ is enhanced, as compared to the single neutron to
proton ratio. The degree of the equilibrium with respect to isospin (isospin
mixing) in terms of the $F_D$ is addressed and its dependence on the symmetry
energy is also discussed.
