is the same as that in [@Yoshida_Eriguchi_2006] and [@Lander_Jones_2009]. In this paper we fix $k=0.1$. Equations (\[Eq:H\_phi\]) and (\[Eq:kappa\]) ensure that the [*toroidal*]{} magnetic field vanishes smoothly at the stellar surface. Incidentally, using these functionals, we obtain the first integral of equation (\[Eq:eular\]) as follows: $$\begin{aligned}
    \int \frac{dp}{\rho} = -\phi_g + \frac{1}{2}(r \sin \theta)^2\Omega^2_0
   + \int \mu(\Psi) \, d\Psi + C \ ,
   \label{Eq:first_int}\end{aligned}$$ where $C$ is an integration constant. The first term of the right-hand side is the gravitational potential. The second term on the right hand side is related to rotation. We can consider it as a rotational potential. Similarly, the third term means the potential of Lorentz force. We can regard this term as the magnetic force potential. Therefore, $\int \mu \, d\Psi$ is considered to be non-force-free contribution from the current density, as is seen in equation (\[Eq:current\]). Since the Lorentz force is given by the cross product $\Vec{j} / c \times \Vec{H}$, the first term of equation (\[Eq:current\]) has no effect on the equation of motion, i.e, it is force-free, and only the second term contributes to the Lorentz force, i.e., non force-free. The distribution of Lorentz force could be changed by adopting different functional 