for many kinds of stationary axisymmetric magnetic field configurations. In fact, as we see from our results, different values for $m$ result in qualitatively different distributions for the magnetic potentials and the [*poloidal*]{} magnetic fields. In other words, we can control the magnetic field distributions to a certain extent by adjusting the value for $m$. This is the reason why we use this functional form of $\mu$ in this paper.

After we choose the functional form of the current density, by using Eq.(\[Eq:current\]) and the definition of the vector potential, we obtain the following partial differential equation of the elliptic type: $$\begin{aligned}
   \Delta (A_\varphi \sin\varphi) 
    = 4 \pi S_A(r,\theta) \sin\varphi \ , \hspace{10pt}
    S_A \equiv - \frac{j_\varphi}{c}.
   \label{Eq:Poisson_A}\end{aligned}$$ As we have seen in the previous paragraph, all the physical quantities related to the vector potential can be expressed solely by $\Psi$. Therefore we need not solve for $A_R$ and $A_z$. It implies that our present formulation does not depend on the gauge condition for the vector potential $\Vec{A}$. Next we impose the boundary conditions for the gravitational potential and the vector potential, chosen as follows: $$\begin{aligned}
   \phi_g \sim {\cal O} \left(\frac{1}{r}\right) \ ,
   \hspace{10pt}(r \rightarrow \infty) \ ,
   \label{Eq:BC_gphi}\end{aligned}$$ $$\begin{aligned}
   