Given a magnetic field $\mathbf{B}=B_{0}\left(\dfrac{xz}{z_{s}^{2}},\dfrac{yz}{z_{s}^{2}}, 1-2\left( \dfrac{x^{2}}{r_{s}^{2}}+\dfrac{y^{2}}{r_{s}^{2}}\right) -\dfrac{z^{2}}{z_{s}^{2}}\right)$ with the perturbation $\Delta \mathbf{B}=\alpha B_{0}\left( 0,kz,ky\right)$. So the goal now is to find the flux function upon the perturbation, which is constant throughout the field line. I read a paper by T. Ahsan and S. A. Cohen (see https://pubs.aip.org/aip/pop/article/29/7/072507/2844219/An-analytical-approach-to-evaluating-magnetic) and it says $\psi =\dfrac{B_{0}}{2}\mathbf{u}^{2}\left( J-\mathbf{v}^{2}\right)$, where $\mathbf{u}=\left( x,y-\alpha k_{z}s^{2},0\right)$ and $\mathbf{v}=\left( \dfrac{x}{r_{s}},\dfrac{y+\dfrac{1}{3}\alpha k\left( z_{s}^{2}+r_{s}^{2}\right) }{r_{s}},\dfrac{z}{z_{s}}\right)$. Yet I cannot approach this answer.
I tried following the substitution in the paper and solved the ODEs, which yielded $3z^{2}=-9\alpha ^{2}-10\alpha c_{2}+ce^{-2s}-3e^{2s}\left( c_{1}^{2}+c_{2}^{2}\right) +3$, where $C_1=xe^{-s}$ and $C_2=(y-\alpha)e^{-s}$. I'm not sure if this is correct but I cannot proceed to the desired result. Any help is appreciated.
