How can we check if a system of equations accepts an integer solution? I have the following system:
$$
x_1 + y_1 + z_1 = 0
$$
$$
x_2 + y_2 + z_2 = 0
$$
$$
x_1^2+y_1^2+z_1^2 = x_2^2 + y_2^2 + z_2^2 
$$
$$
x_1x_2 + y_1y_2 + z_1z_2 = 0
$$
Geometrically, this is equivalent to find an orthogonal basis composed only of integer values, for a plane whose equation in the 3D space is $ x+y+z=0 $
I tried finding a solution using a python script by testing with numbers up to 50, but couldn't find any. I also tried solving it geometrically. I'm staring to think that there is not an integer solution, but I'm not sure how it could be proven. Thanks in advance for sharing your thoughts.

Let $x_1=a$, $y_1=b$, $z_1=c$, $x_2=x$, $y_2=y$ and $z_2=z$.
Thus, $$a+b+c=x+y+z=0,$$ $$a^2+b^2+c^2=x^2+y^2+z^2$$ and $$ax+by+cz=0.$$
Hence, $$ax+by+(a+b)(x+y)=0,$$ which gives
$$(2a+b)x+(a+2b)y=0$$ and since
$$a^2+b^2+(a+b)^2=x^2+y^2+(x+y)^2,$$ we obtain:
$$a^2+ab+b^2=x^2+xy+y^2,$$ which gives
$$(a^2+ab+b^2)(2a+b)^2=y^2((a+2b)^2-(a+2b)(2a+b)+(2a+b)^2)$$ or
$$(a^2+ab+b^2)(2a+b)^2=3y^2(a^2+ab+b^2).$$
Can you end it now?
