By [@DIZ97], it follows that $$\label{eq:Yexpli}
Y(z)=I-\int_{-s}^s\frac{{\mathbf}{F}(w){\mathbf}{h}^t(w)}{w-z}{\,\mathrm{d}}w$$ and $$\label{def:FH2}
{\mathbf}{F}(z)=Y(z){\mathbf}{f}(z), \qquad {\mathbf}{H}(z)=(Y^t(z))^{-1}{\mathbf}{h}(z).$$

Recall the RH problem \[rhp: Pearcey\] for $\Psi$, we make the following undressing transformation to arrive at an RH problem with constant jumps. To proceed, the four rays $\Sigma_k$, $k=1,2,4,5$, emanating from the origin are replaced by their parallel lines emanating from some special points on the real line. More precisely, we replace $\Sigma_1$ and $\Sigma_5$ by their parallel rays $\Sigma_1^{(s)}$ and $\Sigma_5^{(s)}$ emanating from the point $s$, replace $\Sigma_2$ and $\Sigma_4$ by their parallel rays $\Sigma_2^{(s)}$ and $\Sigma_4^{(s)}$ emanating from the point $-s$. Furthermore, these rays, together with the real axis, divide the complex plane into six regions $\texttt{I-VI}$, as illustrated in Figure \[fig:X\].

(100,70)(-5,2) (25,40)[(-1,0)[30]{}]{} (55,40)[(1,0)[30]{}]{} (25,40)(55,40) (25,40)[(-1,-1)[25]{}]{} (25,40)[(-1,1)[25]{}]{}

(55,40)[(1,1)[25]{}]{} (55,40)[(1,-1)[25]{}]{}

(15,40)[(1,0)[1]{}]{} (65,40)[(1,0)[1]{}]{}

(10,55)[(1,-1)[1]{}]{} (10,25)[(1,1)[1]{}]{} (70,25)[(1,-1)[1]{}]{} (70,55)[(1,1)[1]{}]{}

(39,36.3)[$0$]{} (-2,11)[$\Sigma_4^{(s)}$]{}

(-2,67)[$\Sigma_2^{(s)}$]{} (3,42)[$\Sigma_3^{(s)}$]{} (80,11)[$\Sigma_5^{(s)}$]{} (80,67)[$\Sigma_1^{(s)}$]{} (73,42)[$\Sigma_0^{(s)}$]{}

(10,46)[$\texttt{III}$]{} (10,34)[$\texttt{IV}$]{} (68,46)[$\texttt{I}$]{} (68,34)[$\texttt{VI}$]{} (38,55)[$\texttt{II}$]{} (38,20)[$\texttt{V}$]{} (40,40) (25,40) (55,40)

(24,36.3)[$-s$]{} (54,36.3)[$s$]{}

We now define $$\begin{aligned}
\label{eq:YtoX}
 X(z) = \left\{
       \begin{array}{ll}
         Y(z)\Psi(z), & \hbox{for $z$ in the region $\texttt{I}\cup \texttt{III}\cup \texttt{IV} \cup \texttt{VI}$,} \\
         Y(z)\widetilde \Psi(z), & \hbox{for $z$ in the region $\texttt{II}$,} \\
         Y(z)\widetilde \Psi(z)
\begin{pmatrix}
1 & -1 & -1
\\
0 & 1 & 0
\\
0 & 0 & 