We have numerically explored the stable planetary geometry for the multiple systems involved in a 2:1 mean motion resonance, and herein we mainly study the HD 82943 system by employing two sets of the orbital parameters (Mayor et al. 2004; Ji et al. 2004). In the simulations, we find that all stable orbits are related to the 2:1 resonance that can help to remain the semi-major axes for two companions almost unaltered over the secular evolution for $10^{8}$ yr. In addition, we also show that there exist three possible stable configurations:(1) Type I, only $theta_{1} approx 0^{circ}$, (2) Type II, $theta_{1}approxtheta_{2}approxtheta_{3}approx 0^{circ}$ (aligned case), and (3) Type III, $theta_{1}approx 180^{circ}$, $theta_{2}approx0^{circ}$, $theta_{3}approx180^{circ}$ (antialigned case), where two resonant arguments are $theta_{1} = lambda_{1} - 2lambda _{2} + varpi_{1}$ and $theta_{2} = lambda_{1} - 2lambda_{2} + varpi_{2}$, the relative apsidal longitudes $theta_{3} = varpi_{1}-varpi_{2}=Deltavarpi$. And we find that other 2:1 resonant systems (e.g., GJ 876) may possess one of three stable orbits in their realistic motions. Moreover, we also study the existence of the assumed terrestrial bodies at $sim 1$ AU for HD 82943 and GJ 876 systems (see main texts).
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