A^{(i)}\eql {a_i\over y}\,dx\,,\qquad i=1,2,3\,.$$ Finally, the scalar fields, $\alpha(r)$, $\beta(r)$, $\chi(r)$ and $\theta(r)$, depend only on the radial coordinate.

With this Ansatz at hand one can derive a system of BPS equations directly from the supersymmetry variations of the $\mathcal{N}=8$, $d=5$ supergravity. In the next section we outline the calculation and summarize the results.

The BPS equations {#BPSeqs}
-----------------

The supersymmetry variations of $\cN=8$, $d=5$ gauged supergravity read:[^9] $$\begin{aligned}
\label{var32}
\delta\psi_{\mu\,a} & \eql D_\mu\epsilon_a-{1\over 6}\,g\,W_{ab}\gamma_\mu\epsilon^b-{1\over 6}\,H_{\nu\rho\,ab}(\gamma^{\nu\rho}\gamma_\mu+2\gamma^\nu\delta^\rho{}_\mu)\, \epsilon^b\,,\\[6 pt]
\delta\chi_{abc} & \eql \sqrt 2\Big[\gamma^\mu P_{\mu\,abcd}\,\epsilon^d-{1\over 2}\,g A_{dabc}\,\epsilon^d-{3\over 4}\,\gamma^{\mu\nu} H_{\mu\nu[ab}\,\epsilon_{c]|}\Big]\,.
\label{var12}\end{aligned}$$ Under $U(1)_R\times U(1)_F$, the eight gravitini, $\psi^a$, and the supersymmetry parameters, $\epsilon^a$, transform with the charges $$\label{branch8}
\bfs 8\quad \longrightarrow \quad (0,0)+(0,0)+(1,0)+(-1,0)+(\coeff 1 2,\coeff 1 2)+(-\coeff 1 2,-\coeff 1 2)+(\coeff 1 2,-\coeff 1 2)+(-\coeff 1 2, \coeff 1 2)\,.$$ In the following, we are interested in the sector where the gravitini and the corresponding supersymmetries have the unit $R$-charge and are invariant under the flavor symmetry. Those supersymmetry parameters are given by (see (3.1) in [@Khavaev:2000gb] and (3.36) in [@Bobev:2010de]), $$\label{susypar}
\epsilon^a\eql \varepsilon_{(1)}\,\eta^a_{(1)}+\varepsilon_{(2)}\,\eta^a_{(2)}\,,\qquad \epsilon_a\eql \Omega_{ab}\,\epsilon^b\,,$$ where $\varepsilon_{(1)}$ and $\varepsilon_{(2)}$ are a symplectic pair of five-dimensional spinors, $\Omega_{ab}$ is an $8\times 8$ symplectic matrix, and $$\label{}
\eta_{(1)}\eql (1,\, 0,-1,\, 0,\,  0,\,  1,\,  