Poincaré Lemma implies that for any vector field $A\in \mathbb{R}^3$ satisfying $\nabla \cdot A = \text{div }A = 0$, we can express locally this vector field as $A = \nabla \times B = \text{rot }B$, where $B$ is another vector field.
Can we say something similar with respect to a symmetric 2-tensor field $\Sigma$ that satisfies $\nabla\cdot\Sigma = 0$?
Note: $\nabla\cdot\Sigma = \sum_{k=1}^3 \cfrac{\partial \Sigma_{ik}}{\partial x^k}$