the Riemann-Hilbert analysis. As in [@BKext2] and [@Lysov], a Riemann surface of the form (\[eq:curve20\]), together with the zero set of a real function $h(x)$ related to this Riemann surface are essential to the implementation of the Riemann-Hilbert analysis. This function $h(x)$ is given as follows. If we express the solutions $\xi$ in (\[eq:curve20\]) as analytic functions of $z$, then the three solutions to (\[eq:curve20\]) behave as follows when $z\rightarrow\infty$. $$\label{eq:xiinfty}
\begin{split}
\xi_1(z)&=-\frac{1}{z}+O(z^{-2}), \quad z\rightarrow\infty,\\
\xi_2(z)&=-1+\frac{c(1-\beta)}{z}+O(z^{-2}), \quad z\rightarrow\infty,\\
\xi_3(z)&=-\frac{1}{a}+\frac{c\beta}{z}+O(z^{-2}), \quad
z\rightarrow\infty.
\end{split}$$ Then the function $h(x)$ is defined by $$\label{eq:hx}
h(x)=\mathrm{Re}\left(\int_{\lambda_3}^x\xi_2(z)-\xi_3(z)dz\right)$$ In order to implement the Riemann-Hilbert analysis, we must determine the sheet structure of the Riemann surface (\[eq:curve20\]) and the topology of the zero set $\frak{H}$ of $h(x)$. Since our model depends on three parameters $c$, $a$ and $\beta$, while the models in [@BKext2] and [@Lysov] depend only on one parameter $a$, the determination of both the sheet structure of the Riemann surface and the topology of $\frak{H}$ are considerably more difficult in our case and a large part of this paper is devoted to resolve these difficulties so that Riemann-Hilbert analysis like those in [@BKext2] and [@Lysov] can be applied.

To determine the sheet structure of (\[eq:curve20\]), note 