BPS equation in the dual hypermultiplet formulation. Using , , and the fact that $M=N^t N$, we find for the Lagrange multipliers $${\mathrm{d}}\lambda = \pm\, {\mathrm{d}}Y\ ,$$ such that, up to a constant, the solutions for the two extra scalars are completely determined in terms of $\phi$ and $\chi$.

Solutions and instanton action
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We can solve the BPS condition for the three possible matrices $O$. For $O_1=\pm 1$, the condition reads $$\label{O1_BPS}
  * H = \pm \begin{pmatrix} \chi {\overset{\leftrightarrow}{{\mathrm{d}}}} {\mathrm{e}}^{-\phi} \\[2pt] {\mathrm{d}}{\mathrm{e}}^{-\phi} \end{pmatrix}\ .$$ Applying ${\mathrm{d}}*$ to the equation and using the Bianchi identities of $H$, we find that ${\mathrm{e}}^{-\phi}$ must be harmonic, and from the first component it then follows that also $\chi$ satisfies the Laplace equation, $$O_1:\quad {\mathrm{d}}*\! {\mathrm{d}}{\mathrm{e}}^{-\phi} = 0\ ,\quad {\mathrm{d}}*\! {\mathrm{d}}\chi = 0\ .$$ As mentioned above, scalars satisfying these conditions will also solve their field equations.

In the following, we consider for simplicity spherically symmetric configurations (single instantons) in flat space only. The dilaton equation of motion is then solved by $$\label{O1_dil}
  {\mathrm{e}}^{-\phi} = {\mathrm{e}}^{-\phi_\infty} + {\frac{\raisebox{-2pt}{$|Q^{(2)}|$}}{\Omega_D\,
  r^{D-2}}}\ .$$ Here $\Omega_D=(D-2)\mathrm{Vol}(S^{D-1})$, and we have chosen the location of the instanton ($\vec{x}_1$ in ) as the origin. The integration constant $Q^{(2)}$ appearing in the 