Zero-curvature formulation of the Camassa-Holm hierarchy

Question

In the book of Gesztesy and Holden (see the following article of the same authors), they state that the Camassa-Holm hierarchy may be casted as a zero-curvature equation \begin{align} -V_{n,x}+\left[U,V_n\right]=0\,, \end{align} where \begin{align} U=\begin{pmatrix} -1 & 1\\ \frac{4u-u_{xx}}{z} & 1 \end{pmatrix}\,,\quad V_n(z)=\begin{pmatrix} -G_n(z) & F_n(z)\\ \frac{H_n(z)}{z} & G_n(z) \end{pmatrix}\,, \end{align} with $z$ an arbitrary constant and $F_n$, $G_n$ and $H_n$ are functions that has to be specified. Replacing the latter matrices into the zero-curvature equation, one readily obtains the following equations \begin{align} F_{n,x}&=2(G_n-F_n)\,,\\ zG_{n,x}&=(4u-u_{xx})F_n-H_n\,,\\ H_{n,x}&=2H_n-2(4u-u_{xx})G_n\,. \end{align} The authors say that if we make the following polynomial ansatz with respect to $z$, \begin{align} F_n(z)=\sum_{l=0}^n f_{n-l}z^l\,,\quad G_n(z)=\sum_{l=0}^n g_{n-l}z^l\,,\quad H_n(z)=\sum_{l=0}^n h_{n-l}z^l\,, \end{align} it is possible to obtain the following recursive relation for $f_l$ \begin{align} f_0=1\,,\quad f_{l,x}=-2\left(4-\partial_{xx}\right)^{-1}\left[2(4u-u_{xx})f_{l-1,x}+(4u_x-u_{xxx})f_{l-1}\right]\,,\quad l\in\mathbb{N}\,, \end{align} and for $g_l$ and $h_l$, \begin{align} g_l&=f_l+\frac{1}{2}f_{l,x}\,,\quad l\in\mathbb{N}_0\,,\\ h_l&=(4u-u_{xx})f_l-g_{l+1,x}\,,\quad l\in\mathbb{N}_0\,. \end{align} Additionally, for fixed $n$ they obtain $h_n=(4u-u_{xx})f_n$ and \begin{align} h_{n,x}-2h_n+2(4u-u_{xx})g_n=0\,,\quad n\in\mathbb{N}_0\,. \end{align} I obtained all of the previous relations, but I am not being able to obtain the one for $f_{l,x}$, namely, $f_{l,x}=-2\left(4-\partial_{xx}\right)^{-1}\left[2(4u-u_{xx})f_{l-1,x}+(4u_x-u_{xxx})f_{l-1}\right]$.