= Lax pair for the Bogomolny equations
{title2=$[L_0(\lambda),L_1(\lambda)]=0$}
For $D_i=\partial_i+A_i$ and Higgs multiplication $\Phi$, set
$$
L_0=D_1+iD_2-\lambda(D_3+i\Phi),\qquad L_1=D_3-i\Phi+\lambda(D_1-iD_2).
$$
Write $a=F_{12}-D_3\Phi$, $b=F_{31}-D_2\Phi$, $c=F_{23}-D_1\Phi$, 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>.
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