Let \((\textbf{u},\textbf{B})\) be an axisymmetric self-similar solution to the stationary MHD equations with magnetic diffusion, of the form \(\textbf{u}=u^r(r,z)\textbf{e}_{r}+u^{\theta }(r,z)\textbf{e}_{\theta }+u^z(r,z)\textbf{e}_{z}\) and \(\textbf{B}=B^{\theta }(r,z)\textbf{e}_{\theta }\) in cylindrical coordinates \((r,\theta ,z)\) , where \((\textbf{e}_r,\textbf{e}_\theta ,\textbf{e}_z)\) is the orthonormal basis. Under the assumption that \(u^r < \frac{1}{3r} + \frac{2r}{3}\) on the unit sphere and on its intersection with the half-space, respectively, we prove two main results. First, for the domain \(\mathbb {R}^3\setminus \{0\}\) , the velocity field \(\textbf{u}\) must be a Landau solution and the magnetic field \(\textbf{B} \equiv 0\) . Second, in the half-space \(\mathbb {R}^3_+\) with either the no-slip or Navier slip boundary condition, we establish that all such axisymmetric self-similar solutions are trivial, i. e., \(\textbf{u}=\textbf{B}=0\) .
