VN-TST 2004 Problem 2

Find all real values of $\alpha$ , for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation $$f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y$$ for all $x, y \in \mathbb{R}$ .