<s>[INST] Is there any solution to the following system of equations? Is there any solution to the following system of diophantine equations?
$$
\left\{\begin{array}{l} 
2.a^2 = b^2+c^2+d^2 \\
a^2 = e^2+f^2+g^2 , & \mbox{with }((a,b,c,d,e,f,g)>2)\in N\mbox{ and differents among them}
\end{array}
\right.
$$
 [/INST] In the system of equations:
$\left\{\begin{aligned}&X^2+Y^2+Z^2=2R^2\\&F^2+G^2+T^2=R^2\end{aligned}\right.$
Solutions have the form:
$X=4(c^2+f^2+k^2)k^2-(2c^2-2d^2+f^2)f^2-(d^2-c^2)^2$
$Y=4kd(2k^2+d^2-c^2-f^2)$
$Z=4(f^2+c^2-k^2)k^2+(f^2+2c^2-2d^2)f^2+(d^2-c^2)^2$
$F=4kc(2k^2+d^2-c^2-f^2)$
$G=4kf(2k^2+d^2-c^2-f^2)$
$T=(2k^2+d^2)^2+(c^2+f^2)^2-2(d^2+4k^2)(c^2+f^2)$
$R=4(k^2+d^2)k^2+(2c^2+f^2-2d^2)f^2+(d^2-c^2)^2$
$k,c,d,f$ - integers and sets us.
</s>