I do have an equation system of the form: $$\vec M = a \cdot \mathrm{diag}\left(c_1, c_2, c_3. c_4, c_5\right) \cdot\left(\begin{matrix}x^{-2}\\ x^{-1}\\1\\x\\x^2\end{matrix}\right) +b\cdot \mathrm{diag}\left(d_1, d_2, d_3, d_4, d_5\right) \cdot\left(\begin{matrix}y^{-2}\\ y^{-1}\\ 1\\ y\\ y^2 \end{matrix}\right)$$
I want to obtain $x$ and $y$.
$a$ and $b$ are unknown but need not be determined. $\vec M \in \mathbb{R}^5$ is known as well as $c_i \in \mathbb{R}$ and $d_i \in \mathbb{R}$ with $i \in \{1,2,\dots,5\}$. $\mathrm{diag}(c_i)$ denotes a symmetric diagonal matrix with diagonal coeffcients $c_i$.
The equation systems seem to be generic enough so that I suppose someone has already investigated it. Any clues where to look would be greatly appreciated.