For the system of equations:
$$\left\{\begin{aligned}&x^2+y^2+z^2=u^2+w^2+v^2\\&x+y+v=u+w+z\end{aligned}\right.$$
You can record solutions:
$$x=s^2+kt-ks+kq+ts-tq-qs$$
$$y=s^2+kt+ks-kq-ts+tq-qs$$
$$z=s^2+kt+ks-kq+ts-tq+2q^2-3qs$$
$$u=s^2+kt+ks-kq+ts-tq-qs$$
$$w=kt-s^2-ks+kq-ts+tq-2q^2+3qs$$
$$v=kt-s^2+ks-kq+ts-tq+qs$$
$s,q,k,t$ - integers asked us.
For not a lot of other systems of equations:
$$\left\{\begin{aligned}&x^2+y^2+z^2=u^2+w^2+v^2\\&x+y+z=u+w+v\end{aligned}\right.$$
Solutions have the form:
$$x=2s^2+(2q+2t+k)s+kt+qk+2qt$$
$$y=s^2+(q+k)s+qk+kt-t^2$$
$$z=s^2+(t+k)s+qk+kt-q^2$$
$$u=s^2+(q+t+k)s+qt+qk+kt$$
$$w=2s^2+(2q+2t+k)s+qt+qk+kt$$
$$v=s^2+ks+qk+kt-q^2-t^2$$