= Translated conjugacy of a Diophantine vector field
{title2=$D\chi\,\omega=\omega+w\circ\chi+a$}
For a sufficiently small analytic periodic <vector field> $w$, a near-identity angular <diffeomorphism> $\chi$ and a constant vector $a$ solve $D\chi\,\omega=\omega+w\circ\chi+a$. They satisfy $\chi-\mathrm{id}=O(w)$, $a=O(w)$; if $\langle w\rangle=0$, then $a=O(w^2)$. For the defect $e=\mathcal D_\omega\chi-\omega-w\circ\chi-a$, set $A=D\chi$, choose $\Delta a=\langle A^{-1}\rangle^{-1}\langle A^{-1}e\rangle$, and solve $\mathcal D_\omega v=A^{-1}(\Delta a-e)$ with zero mean. Updating $\chi$ by $Av$ leaves a quadratic defect $(De)v-[w(\chi+Av)-w(\chi)-Dw(\chi)Av]$. The <torus small-divisor estimate> and geometrically decreasing strip losses yield a convergent analytic iteration.
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