\] Let $\rho>e^q$, $0<T<T_{\max}$, $\epsilon>0$ and $\xi_t$ be a perturbed Delaunay potential as in Definition \[defPerturbedDelaunayPotential\]. Let $\Phi_t$ be a holomorphic frame associated to $\xi_t$ for all $t$ via the DPW method. Suppose that the monodromy problem is solved for all $t\in (-T,T)$