Solution to a non-linear simultaneous equation I'm trying to solve the non-linear, two-variable system of equations
\begin{align*}
y-x+x^5-\frac{y^4x}{2(1+x^2)^2}-\frac{x^3}{1+y^2} &= 0 \\
-x-y+y^5-\frac{x^4y}{2(1+y^2)^2}-\frac{y^3}{1+x^2} &= 0
\end{align*}
with $x,y \in \mathbb{R}$. You might notice that the second equation is almost obtained from the first by swapping $x$ and $y$, except for the first term whose signs are opposed. I think the only solution is the origin $x=y=0$, and this is seems to be confirmed numerically on WolframAlpha, but I'm not sure, and I don't know how to prove it. You can also write the system as
\begin{align*}
y-x+x^5+\frac{1}{4}\partial_x\left(\frac{y^4}{1+x^2}-\frac{x^4}{1+y^2}\right) &= 0 \\
-x-y+y^5-\frac{1}{4}\partial_y\left(\frac{y^4}{1+x^2}-\frac{x^4}{1+y^2}\right) &= 0
\end{align*}
Any hints/ideas?

I converted both equations into polynomials and put them into Singular to find all solutions.
LIB "solve.lib";
ring r=0,(x,y),dp;
poly p1=2x9y2+2x9+4x7y2+2x7-4x5+2x4y3+2x4y-4x3y2-6x3+4x2y3+4x2y-xy6-xy4-2xy2-2x+2y3+2y;
poly p2=-x6y-x4y-2x3y4-4x3y2-2x3+2x2y9+4x2y7-4x2y3-2x2y-2xy4-4xy2-2x+2y9+2y7-4y5-6y3-2y;
ideal i=p1,p2;
def T=solve(i,30);

There are $89$ distinct solutions, but $88$ of them involve a complex number.
This shows that the last solution, $x=y=0$, is the only real solution.
