0,\,  1)
\,,\qquad 
\eta_{(2)}\eql  (0,\, 1,\, 0,\, 1,-1,\,  0,\,  1,\,  0)\,.$$ In this two-dimensional subspace we have, $$\label{}
\Omega_{ab}\,\eta^b_{(i)}\eql \omega_{ij}\eta^a_{(j)}\,,\qquad 
W_{ab}\,\eta^b_{(i)}\eql \cals W\,\eta^a_{(i)}\,,$$ where $\cals W$ is the superpotential and we have defined $\omega_{12}=-\omega_{21}=1$, $\omega_{11}=\omega_{22}=0$.

We start with the spin-3/2 variations with $\epsilon^a$ in . The $r$-dependence of the Killing spinors for the unbroken supersymmetries is determined by the vanishing of the spin-3/2 variation along the radial direction, $$\label{varr}
\partial_r\varepsilon_{(i)}+{1\over 6}\,e^{f-2h} \,\cals H\,\gamma^{234}\varepsilon_{(i)}+\omega_{ij} 
\Big({g\over 6}\,\,e^f \,\cals W\, 
\gamma^2+{1\over 2}\,\sinh^2\chi\,\theta'
\Big)\varepsilon_{(j)}\eql 0\,,$$ where the prime denotes a derivative with respect to $r$. Then, assuming that the Killing spinors do not depend on the $t$, $z$, $x$ and $y$ coordinates, the remaining variations reduce to three algebraic constraints on $\varepsilon_{(1)}$ and $\varepsilon_{(2)}$: $$\begin{aligned}
\label{vartx}
\big(3\,e^{-f}f'\,\gamma^2 - e^{-2h}\,\cals H\,\gamma^{34}\big)\,\varepsilon_{(i)} -{g}\,\omega_{ij}\,\cals W\, \varepsilon_{(j)} & \eql 0\,,\\[6 pt]
\label{varxy}
(3\,e^{-f}h'\,\gamma^2   +2\,e^{-2h}\,\cals H\,\gamma^{34}\big)\,\varepsilon_{(i)}-g\,\omega_{ij}\,\cals W\,\varepsilon_{(j)}& \eql 0\,,\\[6 pt]
\label{varxydef}
   2\, \gamma^4\,\varepsilon_{(i)}-{g }\,\omega_{ij}\, \, \Lambda\, \gamma^3\varepsilon_{(j)}& \eql 0\,.\end{aligned}$$ Here and above, we have defined $$\label{}
\begin{split}
\cals H  & \eql  e^{2\alpha-2\beta}a_1+e^{2\alpha+2\beta}a_2+e^{-4\alpha} a_3  \,,\\[6 pt]
\Lambda & \eql a_1+a_2+3 a_3+(a_1+a_2-a_3)\cosh 2\chi\,.
\end{split}$$ The first equation, , arises from the variations along $\mathbb{R}^{1,1}$, while the remaining two, and , from the variations along the Riemann surface. The function $\cals H$ 