of the magnetic flux function $\Psi$. Notice, in particular, that the [*toroidal*]{} component of magnetic field is given as $$H_\varphi = \frac{\kappa (\Psi)}{r \sin \theta}
\label{Eq:H_phi}$$ which can be derived from equations (\[Eq:eular\]), (\[Eq:rotH\]) and (\[Eq:current\]). It should be noted that these two arbitrary functions are conserved along the [*poloidal*]{} magnetic field lines. Although the meanings of these two functions are described in previous works (see, e.g., @Lovelace_et_al_1986), in this paper we will explain their meanings differently from our point of view.

Since we have assumed that there is no electric current in the vacuum region, in other words that there is no [*toroidal*]{} magnetic field outside the star (see equation \[Eq:current\]), the form for $\kappa$ needs to be a special one. The simplest form can be $\kappa=$ constant (@Ioka_Sasaki_2004, @Haskell_Samuelsson_Glampedakis_2008), but for this choice of $\kappa$ the [*toroidal*]{} magnetic field would extend to the vacuum region. In order to avoid this possibility, we choose the functional form of $\kappa$ as follows: $$\begin{aligned}
  \kappa (\Psi) =
   \left\{
     \begin{array}{lr}
        0 \ , & \mathrm{for} \hspace{10pt} \Psi \leq \Psi_{\max} \ , \\
        \dfrac{\kappa_0}{k+1}(\Psi - \Psi_{\max})^{k+1} \ , 
         & \mathrm{for} \hspace{10pt} \Psi \geq \Psi_{\max} \ ,
     \end{array}
   \right.
   \label{Eq:kappa}\end{aligned}$$ This choice of $\kappa$ 