and $\xi_3(z)$. This determines the branch structure of the Riemann surface ${\mathcal{L}}$. The branch cut of $\xi_2(z)$ and $\xi_3(z)$ will eventually be chosen to be a contour that goes from $\lambda_4$ to $\lambda_3$ intersecting the real axis at a point in $(\lambda_1,\lambda_2)$, but in the next section it will be chosen in a few different ways according to the situation.

### Geometry of the problem {#se:geo}

As explained in the introduction, the determination of the zero set of $h(x)$ in (\[eq:hx\]) is considerably more difficult than it is in [@BKext2] and [@Lysov]. In this section we will carry out a thorough analysis of this set and determine its topology.

For the implementation of the Riemann-Hilbert analysis, it is more convenient to consider a measure $\hat{F}_{N}$ instead of the measure $\underline{F}$.

Let $c_N=\frac{N}{M}$ and $H_N(t)$ be the e.d.f $$dH_N(t)=(1-\beta_N)\delta_1+\beta_N\delta_a.$$ where $\beta_N=\frac{N_1}{N}$.

Then $\hat{F}_N$ is the measure whose Stieltjes transform $m_N(z)=m_{\hat{F}_N}(z)$ is the unique solution of $$\label{eq:mN}
z(m_N)=-\frac{1}{m_N}+c_N\int_{\mathbb{R}}\frac{t}{1+tm_N}dH_N(t),$$ in $\mathbb{C}^+$ that behaves like $-\frac{1}{z}$ as $z\rightarrow\infty$. Note that, as pointed out in [@Baikspike], the measure $\hat{F}_N$ is not the eigenvalue distribution for finite $N$, instead, it is only defined through the equation (\[eq:mN\]). From (\[eq:mN\]), we see that $m_N(z)$ is the solution 