Considering this system of integral equations, where $\gamma \in \mathbb{R} $ and $\alpha\in \mathbb{C}$ are the unknown to solve :
$$ 1=\int p(u) \frac{ -1}{\gamma-\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}$$$$ 1=\int_{-\infty}^{\infty} p(u) \frac{ -1}{\gamma-\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}\mathrm{d}u$$
$$\alpha^*=\int p(u) \frac{ u-z+\tau \alpha}{\gamma-\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}$$$$\alpha^*=\int_{-\infty}^{\infty}p(u) \frac{ u-z+\tau \alpha}{\gamma-\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}\mathrm{d}u$$
$$\alpha=\int p(u) \frac{ u-z^*+\tau \alpha^*}{\gamma-\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}$$$$\alpha=\int_{-\infty}^{\infty}p(u) \frac{ u-z^*+\tau \alpha^*}{\gamma-\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}\mathrm{d}u$$
Where $\alpha^*$represents the complex conjugate of $\alpha$ and $p(u)$ is a probability distribution. Is it possible to solve these equations without further information on $p(u)$? Or is it at least possible to change their representation to make them more friendly ?
The solution $\alpha$ would then give an equation representing the boundaries of the eigenvalues of a random matrix :
$$ 1=\int p(u) \frac{ 1}{\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}$$$$ 1=\int_{-\infty}^{\infty}p(u) \frac{ 1}{\left(u-z^{*}+\tau \alpha^*\right)\left(u-z+\tau \alpha\right)}\mathrm{d}u$$
Therefore I see $\gamma$ and $\alpha$ as intermediate steps, the boundaries only depend directly on $p(u)$ and not $\alpha$ strictly speaking.
Any remark, reference or advice is always very appreciated. Thank you.