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I think the following for the lowest energy distribution of the dumbbell:
In case α>0, the configuration with the dumbbell axis perpendicular on the plane of the rings (along the z-axis) is minimal.
In case α=0, all configurations have minimal energy.
In case α<0, the configurations with the dumbbell axis parallel to the plane of the rings are minimal. The axis lies thus in the xy-plane.
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