Could someone provide a hint as to why $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$
where $b$ is a constant, $i$ is $\sqrt {-1}$, implies that $$2\int d^3x \,\,x_ia_j(\vec x)=\epsilon_{ijk} \Big[\int \,\,d^3x \,\, \vec x\times \vec a(\vec x)\Big]_k-ib \int\,\,d^3x\,\, x_ix_jc(\vec x)$$?
Context/Interpretation: $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$ is obtained from $$\nabla \cdot [\vec a(\vec x) \exp(-ibt)]= {\partial\over\partial t}[c(\vec x)\exp(-ibt)]$$ which can be interpreted as a conservation law. I don't have any more context information...