the asymptotic conserved mass and angular momenta. Since we are concerned with stationary, axisymmetric spacetimes with Killing symmetries $\xi_I$, the conserved charges $M_{ADM}$ and $J_a$ are defined, respectively, by $$\begin{aligned}
 M_{ADM} &=& -\frac{3}{32\pi} 
        \int_{S^3_\infty}
                 dS^{\mu \nu} \nabla_\mu (\xi_3)_\nu \,,  
\label{def:Mass} 
\\
 J_a &=& \frac{1}{16\pi}\int_{S^3_\infty} 
               dS^{\mu \nu} \nabla_\mu (\xi_a)_\nu \,. 
\label{def:Ja} \end{aligned}$$ We write below our boundary conditions for $\Phi^A$ in terms of the Weyl-Papapetrou coordinates. Therefore, in particular, relevant conditions at infinity—see below eqs. (\[gtt\]) – (\[grhorho\])—are derived from the above fall-off behavior, eq. (\[condi:AF\]), by the coordinate transformation $$\begin{aligned}
 \rho = \frac{1}{2} r^2 \sin 2\theta \,, \quad 
    z = \frac{1}{2} r^2 \cos 2\theta \,. \end{aligned}$$ Then, we can find that the boundary conditions given in this section are, in fact, the same as the limiting behavior of $\Phi^A$ for the exact solution of Cvetič-Youm [@CY96] at the corresponding boundaries, which we discuss in Appendix \[sec:solution\].

In terms of the Weyl-Papapetrou coordinate system introduced in Sec. \[subsec:II-B\] and the rod-structure [@Harmark], the boundary $\partial \Sigma$ of the base space $\Sigma=\{(\rho,z)|\ \rho>0,\ -\infty<z<\infty \}$ is described as a set of three rods and the infinity: Namely,

1.  the $\phi$-invariant plane: $\partial \Sigma_\phi=\{(\rho,z)|\rho=0,k^2<z<\infty \}$ with the rod vector $v=(0,1,0)$ ,

2.  the 