degeneracy. It is easily verified that field configurations satisfying have vanishing energy-momentum tensors, hence they can exist only in Ricci-flat spaces. We therefore have to amend our BPS condition by the equation $R_{\mu\nu}(g)=0$.

For the field equations of the tensors, ${\mathrm{d}}(M\!*\!H)=0$, to hold we must have $$\label{closed}
  {\mathrm{d}}(N^t O E) = 0\ .$$ This condition also guarantees that the topological term in is closed and hence can locally be written as a total derivative. As a consequence, it does not contribute to the equations of motion such that also the field equations for the scalars are guaranteed to be satisfied. The latter follow from requiring that the solution of correspond to closed forms for $H^a$, $${\mathrm{d}}(N^{-1} O *\! E)=0\ .$$

To determine the O(2) matrices that are compatible with , we parametrize $O$ by $$O = \begin{pmatrix} 1 & 0 \\[2pt] 0 & {\epsilon}\end{pmatrix}
  \begin{pmatrix} c & -s \\[2pt] s & c \end{pmatrix}\ ,$$ where the functions $c(\phi,\chi)$ and $s(\phi,\chi)$ are constrained by $c^2+s^2=1$, and ${\epsilon}=\pm 1$ for the two components of O(2) with $\det O={\epsilon}$. Equation then gives rise to the differential equations $$\begin{aligned}
  0 & = {\partial}_\phi c - {\mathrm{e}}^{\phi/2} {\partial}_\chi s \notag \\*
  0 & = {\partial}_\phi 