i}\int \frac{q_{\mu_n,n-1}(x)}{z-x} \, \mbox{d}\mu_n(x) 
\end{array}\right).$$

The key of our procedure follows the ideas introduced in [@AV]. We find a relationship between $Y$ and the matrix solution $R: \mathbb{C}\setminus \gamma \rightarrow \mathbb{C}^{2\times 2}$ corresponding to another Riemann-Hilbert problem (R-RHP) for a closed Jordan curve $\gamma$ positively oriented surrounding the interval $[-1,\,1]$:

1.  $R \in \mathcal{H} (\mathbb{C}\setminus \gamma)$,

2.  $R_+(\zeta)=R_-(\zeta)V_n(\zeta),$ $\zeta \in \gamma$, with $V_n\in \mathcal{H}(D)$,

3.  $R(z) \rightarrow \mathbb{I}$ as $z \to \infty$,

where $V_n=\mathbb{I}+\mathcal{O}(\varepsilon^n)$ with $0\leq \varepsilon<1$, uniformly on compact subsets of $K$ as $n \to \infty$. Those conditions imply that $R=\mathbb{I}+\mathcal{O}(\varepsilon^n)$ uniformly on $\mathbb{C}$ as $n \to \infty$. There is a chain of transformations to arrive from $Y$ to $R$, which we represent $Y \rightarrow T \rightarrow S \rightarrow R$. Once we have arrived to $R$, we recover the entries of $Y$ going back from $R$ to $Y$.

From [@BLS2 Corollary 4] we have that the zero counting measures $\nu_n$ defined in (\[zerocountingmeasures\]) corresponding to the monic orthogonal polynomials $\displaystyle q_{\mu_n,n}(z)=\prod_{j=1}^n\left(z-x_{j,n}\right)$ with respect to the varying measures $\mu_n$, satisfy $$\label{equilibriummeasure}
\nu_n \overset{\star}{\rightarrow} \overline{\nu}=(1-a) \widetilde{\sigma} +a \lambda_0 \quad \mbox{as} \quad n \to \infty,$$ where $\widetilde{\sigma}$ denotes the balayage of the measure $\sigma$ out of $\mathbb{C}\setminus [-1, \,1]$ onto $[-1,\,1]$.

The 