  In this paper, we investigate the well-posedness and ill-posedness issues for
the incompressible stationary Hall-magnetohydrodynamic (Hall-MHD) system in
$\mathbb{R}^3.$ We first show the existence and uniqueness of solutions
provided with the forces in $\dot B^{3/p-3}_{p,r}(\mathbb{R}^3)$ for $1\leq p
<3$ and $r=1$. Moreover, this result can be extended to any $1\leq r\leq
\infty$ whenever $p=2,$ without any additional assumption on the physical
parameters. On the other hand, we establish some ill-posedness results for
Hall-MHD system by using the discontinuity of the solution mapping of the
three-dimensional stationary Navier-Stokes equations in \emph{critical}
function spaces $\dot{B}^{3/p-1}_{p,r}(\mathbb{R}^3)$ ($p\geq 3$).
