Suppose $u \in \mathbf{R}^n$ satisfies $\|u\|_2 = 1$ and suppose I know that $$ y = x + \left(\sqrt{(x^T u)^2 - x^Tx +1}\right)u $$
If $y, u \in \mathbf{R}^n$ are given, is there a clean way to characterize the solutions $x \in \mathbf{R}^n$ to this equality in terms of $y$ and $u$?