  The effect of $\delta-$ and $\omega-\rho-$meson cross couplings on asymmetry
nuclear systems are analyzed in the frame-work of an effective Field theory
motivated relativistic mean field formalism. The calculations are done on top
of the G2 parameter set, where these contributions are absent. We calculate the
root mean square radius, binding energy, single particle energy (for the
$1^{st}$ and last occupied orbits), density and spin-orbit interaction
potential for some selected nuclei and evaluate the $L_{sym}-$ and $E_{sym}-$
coefficients for nuclear matter as function of $\delta-$ and
$\omega-\rho-$meson coupling strengths. As expected, the influence of these
effects are negligible for symmetry nuclear system and these effects are very
important for systems with large isospin asymmetry.
