DPW potentials defined for $(t,z)\in(-T,T)\times D_{\epsilon}^*$ by $$\begin{aligned}
    \xi_t(z) = A_tz^{-1}dz + C_t(z)dz
    \end{aligned}$$ where $A_t\in\Lambda{\mathfrak{sl}}(2,{\mathbb{C}})_{\rho}$ is a Delaunay residue as in satisfying and $C_t(z)\in{\mathfrak{sl}}(2,{\mathbb{C}})_{\rho}$ is ${\mathcal{C}}^1$ with respect to $(t,z)$, holomorphic with respect to $z$ for all $t$ and satisfies $C_0(z)=0$ for all $z$.

\[theoremPerturbedDelaunay\] Let $\rho>e^q$, $0<T<T_{\max}$, $\epsilon>0$ and $\xi_t$ be a perturbed Delaunay potential ${\mathcal{C}}^2$ with respect to $(t,z)$. Let $\Phi_t$ be a holomorphic frame associated to $\xi_t$ for all $t$ via the DPW method. Suppose that the family of initial conditions $\phi_t$ is ${\mathcal{C}}^2$ with respect to $t$, with $\phi_0=\widetilde z_0^{A_0}$, and that the monodromy problem is solved for all $t\in (-T,T)$. Let $f_t={ \mathrm{Sym}}_q\left({\mathop \mathrm{Uni}}\Phi_t\right)$. Then,

1.  For all $\delta>0$, there exist $0<\epsilon'<\epsilon$, $T'>0$ and $C>0$ such that for all $z\in D_{\epsilon'}^*$ and $t\in(-T',T')\backslash\{0\}$, $$\begin{aligned}
            {d_{{\mathbb{H}}^3}\left(f_t(z),f_t^{\mathcal{D}}(z)\right)}\leq C|t||z|^{1-\delta}
            \end{aligned}$$ where $f_t^{\mathcal{D}}$ is a Delaunay immersion of weight $2\pi t$.

2.  There exist $T'>0$ and $\epsilon'>0$ such that for all $0<t<T'$, $f_t$ is an embedding of $D_{\epsilon'}^*$.

3.  The limit axis as $t$ tends to $0$ of the Delaunay immersion $f_t^{\mathcal{D}}$ oriented towards the end at $z=0$ is given by: $$\frac{1}{\sqrt{2}}\begin{pmatrix}
            1 & -e^q\\
            e^{-q} & 1
            \end{pmatrix} \cdot {\mathrm{geod}}\left({\mathrm{I}}_2,-\sigma_3\right)\quad \text{in the spherical family ($r>s$)},$$ $${\mathrm{geod}}\left({\mathrm{I}}_2,-\sigma_1\right)\quad \text{in 