An analytical solution of the integral equation $ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz $

Question

How to solve for the f(x) in the integral equation?

$$ f(x) = \int_{\mathbb{R} \setminus \{0\}} f(x+z) - f(x)\frac{C_2}{|z|^{1+\alpha}} dz \\ \alpha \in (1,2), C_2\text{ is a positive constant.} \\ \text{What is the definition of the function } f(x) \text{ in the given integral equation?} $$

I have tried using Taylor series method to attempt to obtain some numerical solutions, as well as attempting to use iterative methods to derive some results, but without success.

I am unable to figure out on how to solve it.

Can someone please give some hints?