From my electromagnetism courses I remember that a divergenceless vector field (such as the magnetic field) can be expressed as the curl of another vector field:
$$\boldsymbol{\nabla}\cdot\boldsymbol{b} = \boldsymbol{0} \Leftrightarrow \boldsymbol{b} = \boldsymbol{\nabla} \times \boldsymbol{a}$$
I am wandering if a similar expression exists for second order tensors. More precisely, if I consider a 2D symmetric second-order tensor such as:
$$ \boldsymbol{T} = \begin{bmatrix} a & c \\ c & b \end{bmatrix}$$
and assume that it verifies
$$\boldsymbol{\nabla}\cdot\boldsymbol{T} = \boldsymbol{0}$$
Can I express it as an algebraic combination of the nabla operator and some other vector and/or tensor fields?