follows from eq. (\[eq:deriv\]) that $\Psi$ must be a constant matrix over the region $\Sigma$. Therefore, in particular, if $\Psi$ is shown to be zero on some part of the boundary $\partial \Sigma$, it immediately follows that $\Psi$ must be identically zero over the base space $\Sigma$, implying that the two solutions $M_{[0]}$ and $M_{[1]}$ must coincide with each other. This is indeed the case under our boundary conditions discussed in the next section.

Boundary value problems {#sec:coincidence}
=======================

In this section, we derive necessary boundary conditions for determining the scalar fields $\Phi^A=(\lambda_{ab},\omega_a,\psi_a,\mu)$, requiring asymptotic flatness at infinity, regularity on the two rotation axes (i.e., the $\phi$-invariant plane and the $\psi$-invariant plane), and on the event horizon (of which cross-sections are assumed to be topologically spherical). Note that by [*asymptotically flat*]{}, we mean that the spacetime metric has the following fall off behavior at large distances, $$\begin{aligned}
ds^2 &\simeq& \left(-1+\frac{8M_{ADM}}{3\pi r^2}+{\cal O}(r^{-3})\right)dt^2
            - \left(
                    \frac{8J_\phi\sin^2\theta}{\pi r^2}+{\cal O}(r^{-3})
              \right)dtd\phi\nonumber\\ 
       & &- \left(
                    \frac{8J_\psi\cos^2\theta}{\pi r^2}+{\cal O}(r^{-3})
              \right)dtd\psi 
\nonumber\\
      &   & + \left( 1 + {\cal O}(r^{-1}) \right)
                  \left(dr^2+
                        r^2\left(d\theta^2 + \sin^2\theta d\phi^2  
                                  + \cos^2\theta d\psi^2 \right)
                  \right) \,, 
\label{condi:AF}\end{aligned}$$ having the spherical spatial infinity, $S^3_\infty$. Here the constants $M_{ADM}$ and $J_a$ are 