Tensor fields on the plane and Riesz transforms

In this paper we study symmetric tensor fields via complex coordinate system. The formulas for divergence $\delta$ and symmetric gradient $d$ of tensor fields in complex variables are derived thus we get the equations of Beltrami type. We obtain the general representation for the solenoidal tensor fields on the plane, which involves the Riesz transforms, their powers and the one real generating function $f\in L_2(\mathbb R^2)$. We present also the Helmholtz decomposition of the tensor fields in terms of Riesz transforms.