One system of Diophantine equations.

[Individ]

This system of Diophantine equations was proposed.
Form APMO 2000. 2.
Has this type:

$\left\{\begin{aligned}&x_1+x_2+x_3+x_4=x_4+x_5+x_6+x_7=x_7+x_8+x_9+x_1\\&x_1^2+x_2^2+x_3^2+x_4^2=x_4^2+x_5^2+x_6^2+x_7^2=x_7^2+x_8^2+x_9^2+x_1^2\end{aligned}\right.$

One easy solution can be written as:

$x_1=6s^2-16ks-3as+6ka+8k^2$

$x_2=15s^2-56ks-3as+6ka+52k^2$

$x_3=3s^2-7ks-3as+6ka+2k^2$

$x_4=-3s^2+14ks-3as+6ka-16k^2$

$x_5=6s^2-20ks-3as+6ka+16k^2$

$x_6=3s^2-4ks-3as+6ka-4k^2$

$x_7=15s^2-55ks-3as+6ka+50k^2$

$x_8=3s^2-10ks-3as+6ka+8k^2$

$x_9=-3s^2+16ks-3as+6ka-20k^2$

$s,a,k$ - integers which we ask.

[Individ]

For a system of equations.

$\left\{\begin{aligned}&x_1+x_2+x_3+x_4=x_4+x_5+x_6+x_7=x_7+x_8+x_9+x_1\\&x_1^2+x_2^2+x_3^2+x_4^2=x_4^2+x_5^2+x_6^2+x_7^2=x_7^2+x_8^2+x_9^2+x_1^2\end{aligned}\right.$

It is better to write such a decision.

$x_1=2(a-b)ba^2k^2-2a(3a-2b)bks+2(2a-b)bs^2+Q$

$x_2=(4a^2-3b^2)a^2k^2-4a(2a^2-b^2)ks+(4a^2-b^2)s^2+Q$

$x_3=(2a^2-4ab+b^2)abk^2-(4a^2-5ab+b^2)bks+(2a-b)bs^2+Q$

$x_4=-4a^2(a-b)^2k^2+a(8a^2-16ab+7b^2)ks-(4a^2-8ab+3b^2)s^2+Q$

$x_5=4b(a-b)a^2k^2-2a(4a-3b)bks+2(2a-b)bs^2+Q$

$x_6=-a^2b^2k^2-2a(a-b)bks+(2a-b)bs^2+Q$

$x_7=a(4a^3-4ab^2+b^3)k^2-(8a^3-5ab^2+b^3)ks+(4a^2-b^2)s^2+Q$

$x_8=2(a-b)ba^2k^2-(4a-3b)abks+(2a-b)bs^2+Q$

$x_9=-(4a^2-8ab+5b^2)a^2k^2+8a(a-b)^2ks-(4a^2-8ab+3b^2)s^2+Q$

$Q$ - any integer.