Solve system of non-linear equations $\frac{1}{x^2}-\frac{1}{(y-x)^2}-\frac{1}{(1-x)^2}= \frac{1}{y^2}+\frac{1}{(y-x)^2}-\frac{1}{(1-y)^2}=0$ Whilst solving a problem, the following system of non-linear equations arose:
$$0= \frac{1}{x^2}-\frac{1}{(y-x)^2}-\frac{1}{(1-x)^2}$$
$$0= \frac{1}{y^2}+\frac{1}{(y-x)^2}-\frac{1}{(1-y)^2}$$
I have tried to solve it via brute force and by trying to add them both in a way that could give me the answer, but I still haven't got it.
I can't manage to find the intersection of the two parametric equations that these equations give. I know that I can find approximate solutions with the Wolfram Alpha tool, but I would like to know, if any, an analytical or strong numerical way to proceed.
Note: Out of all the solutions ($8$ it seems, $4$ real and $4$ complex), I only need the one that takes real values for $x$ and $y$, with $0<x<y<1$.

Take the cue of $0<x<y<1$ and observe that $x+y=1$ satisfies both equations
$$\frac{1}{x^2}-\frac{1}{(y-x)^2}-\frac{1}{(1-x)^2}=
\frac{1}{y^2}+\frac{1}{(y-x)^2}-\frac{1}{(1-y)^2}=0$$
which leads to $\frac{1}{x^2}-\frac{1}{(1-2x)^2}-\frac{1}{(1-x)^2}=0$, or
$$x^4+6x^3-11x^2+6x -1=0$$
Factorize
$$[x^2+3(1+\sqrt2)x-(1+\sqrt2)][x^2+3(1-\sqrt2)x-(1-\sqrt2)]=0$$
to obtain the solution in the domain $(0,1)$
$$x= \frac12\left( \sqrt{31+22\sqrt2}-3\sqrt2-3\right)
$$
$$ y= \frac12\left(- \sqrt{31+22\sqrt2}+3\sqrt2+5\right)$$
