Lax pair for the Bogomolny equations ($[L_0(\lambda ),L_1(\lambda )]=0$)

For $D_i={\partial }_i+A_i$ and Higgs multiplication $\Phi$, setWrite $a=F_{12}-D_3\Phi$, $b=F_{31}-D_2\Phi$, $c=F_{23}-D_1\Phi$, where $D_i\Phi ={\partial }_i\Phi +[A_i,\Phi ]$. Direct expansion gives $[L_0,L_1]=-b+ic-2i\lambda a-{\lambda }^2(b+ic)$. Its vanishing for every complex $\lambda$ is therefore equivalent to $a=b=c=0$, the Bogomolny equations. It is a parameter-dependent zero-curvature representation.