The general problem I am trying to solve is the following; is there a non-constant endomorphic function $\eta: \mathbb{R}^n \to \mathbb{R}^n$ such that $$\langle \eta(x_1),\, \eta(x_2) \rangle = \alpha(x_1 - x_2)$$ holds for all $x_1,\, x_2 \in \mathbb{R}^n$, where $\alpha$ is some non-zero function on $\mathbb{R}^n$ and $\langle \cdot, \cdot \rangle$ is the Euclidean inner product.
For the moment I am focusing on the case where $n = 1$, and I have deduced that $\alpha$ must be even (e.g. $\alpha(x) = \beta(x) + \beta(-x)$ for some $\beta$) and that $\eta(x)^2$ must be constant.
I am also aware that if we disregard the endomorphism restriction on $\eta$ and define $\eta: x \mapsto (\cos(x),\, \sin(x))$ then this property is satisfied; in fact if we keep adding more $\cos$ and $\sin$ of arbitrary but equal frequency this still holds (and $\eta$ becomes the Fourier encoding). The challenge seems to be this endomorphism restriction, and I am unsure if there even exists a solution (and how to prove so).