like to show that the boundary integral, eq. (\[eq:integral\]), indeed vanishes under our preferable boundary conditions that require the regularity on the three rods and asymptotic flatness at infinity. For this purpose, in the following we evaluate the limiting behavior of the integrand, $\rho\ \partial_z{\rm tr}\ \Psi$, of eq. (\[eq:integral\]), separately on each boundary (i)–(iv).

\(i) $\phi$-invariant plane: $\partial\Sigma_\phi=\{(\rho,z)|\rho=0,\ k^2<z<\infty\}$. The regularity on the $\phi$-invariant plane requires that for $\rho\to 0$, the scalar fields behave as $$\begin{aligned}
&&\lambda_{\phi\phi}\simeq {\cal O}(\rho^2) \,,\label{eq:a1}\\ 
&&\lambda_{\psi\psi}\simeq {\cal O}(1) \,,\label{eq:a2}\\
&&\lambda_{\phi\psi}\simeq {\cal O}(\rho^2) \,, \label{eq:a3}\\
&&\omega_{\phi}\simeq -\frac{2J_\phi}{\pi}+{\cal O}(\rho^2) \,, \label{eq:a4}\\
&&\omega_{\psi}\simeq -\frac{2J_\psi}{\pi}+{\cal O}(\rho^2) \,,\label{eq:a5} \end{aligned}$$ and $$\begin{aligned}
&&\psi_{\phi}\simeq {\cal O}(\rho^2)\,, \label{eq:a6}\\
&&\psi_{\psi}\simeq {\cal O}(1) \,,\label{eq:a7}\\
&&\mu\simeq \frac{2Q}{\sqrt{3}\pi}+{\cal O}(\rho^2) \,, \label{eq:a8} \end{aligned}$$ where the boundary conditions, eqs. (\[eq:a1\])-(\[eq:a3\]) and eqs. (\[eq:a6\])-(\[eq:a7\]), come from the requirement that $\partial\Sigma_\phi$ is the $\phi$-invariant plane, i.e., the plane invariant under the rotation with respect to the axial Killing vector $\partial/\partial \phi$. The conditions, eqs. (\[eq:a4\])-(\[eq:a5\]), are derived from eq. (\[eq:twist\]). In the derivation of the condition (\[eq:a8\]), eq. (\[eq:charge2\]) is used. Then for two solutions, $M_{[0]}$ and $M_{[1]}$, with the same mass, the same angular momenta, and the same electric charge, $\rho\ {\rm tr}\Psi$ behaves as $$\begin{aligned}
\rho\ \partial_z{\rm tr}\ 