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In this paper, we study vacuum static spaces with the complete divergence ofthe Bach tensor and Weyl tensor. First, we prove that the vanishing of completedivergence of the Bach tensor and Weyl tensor implies the harmonicity of themetric, and we present examples in which these conditions do not imply Bachflatness. As an application, we prove the non-existence of multiple black holesin vacuum static spaces with zero scalar curvature. On the other hand, we provethe Besse conjecture under these conditions, which are weaker than harmonicityor Bach flatness. Moreover, we show a rigidity result for vacuum static spacesand find a sufficient condition for the metric to be Bach-flat.

Vacuum static spaces with vanishing of complete divergence of Bach tensor and Weyl tensor