  In this paper, we consider the solvability of the two-dimensional stationary
Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In [6], it was
proved that the stationary Navier--Stokes equations on $\mathbb{R}^2$ is
ill-posed for solutions around zero. In contrast, considering solutions around
the non-zero constant flow, the perturbed system has a better regularity in the
linear part, which enables us to prove the unique existence of solutions in the
scaling critical spaces of the Besov type.
