Let $p(x)$ be a probability distribution. Without further information is it possible to solve the following equation for the unknown $z\in \mathbb{C}$ ?
$$ z = \int\frac{p(x)}{x-z^*}\mathrm{d}x $$ Where $z^*$ represents the complex conjugate of z.
I tried feeding various methods such as feeding this expression in the Euler-Lagrange equation, but I am starting to believe that without further information on $p(x)$ this equation is the best we can get.
Any remark or advice is always appreciated, thank you very much.