Does there exist any integer solution for the diophantine system of equations below:
$\left\{\begin{array}{l} x^2=a^2+b^2-ab \\ y^2=a^2+b^2+ab \end{array} \right. ,$
Notice: I know that the two equations above are two similar ellipses with $x=y$ axis that transform to eachother by $\frac {\pi}{2}$ rotation angle, but unfortunately I'm not familiar enough with algebric geometry to solve this system.
Here for example I've find a solution for the first equation for $(x,a,b)=(7,8,3)$ by changing it to a simpler form $m(m+k)=n(n+2k)$ where $(x,a,b)=(n+k,m+k,m)$, but this solution doesn't hold in the second equation.
Now it is clear that we are searching for integer solutions of the intersection points of these two ellipses.
$$ \left( x^2 + xy + y^2 \right) \left( x^2 - xy + y^2 \right) = x^4 + x^2 y^2 + y^4 $$
Page 19 In Mordell's book Diophantine Equations includes a proof that the only solutions for $$ x^4 + x^2 y^2 + y^4 = z^2 \; , \; \gcd(x,y) = 1 $$ integers have either $x=0$ or $y=0$ and the other $\pm 1.$ This is formula (7'), given as an example of (7), that being
$$ x^4 + kx^2 y^2 + y^4 = z^2 \; , \; \gcd(x,y) = 1 $$ Chapter 4 is called Quartic Equations with only Trivial Solutions