solutions to special diophantine equations Let $0\le x,y,z,u,v,w\le n$ be integer numbers obeying
\begin{align*}
x^2+y^2+z^2=&u^2+v^2+w^2\\
x+y+v=&u+w+z\\
x\neq& w
\end{align*}
(Please note that the second equality is $x+y+v=u+w+z$ NOT $x+y+z=u+v+w$. This has lead to some mistakes in some of the answers below)
How can the solutions to the above equations be characterized. One class of solutions is as follows
\begin{align*}
x=&u\\
y=&w\\
z=&v
\end{align*}
 A: This system is well studied. You can find full description of solutions in 
"Introduction to the theory of numbers" by Leonard E. Dickson. (See Theorem 47). 
If all the variables are between $1$ and $P$ then the number of solutions is
$$\frac{18}{\pi^2}P^3\log P+O(P^3),$$ 
see "An asymptotic formula for the number of solutions of a system of equations"
N.N. Rogovskaya - Diophantine approximations, Part II (Russian), 1986. She used more natural parametrization: if $a_1=x-u$, $a_2=y-v$, $a_3=z-w$, $b_1=x+u$, $b_2=y+v$, $b_3=z+w$, then the system is equivalent to
$$a_1+a_2+a_3=0,\qquad a_1b_1+a_2b_2+a_3b_3=0.$$ All solutions of the equation
$$kb_1+lb_2-(k+l)b_3=0$$
for $(k,l)=1$ are
$$b_1=b_3-lt,\quad b_2=b_3+kt\quad(b_3,t\in\mathbb{Z}).$$
Your system can be also studied with circle method. And this is unique case (among Vinogradov's type systems) when trigonometric integal is known explicitly (from Rogovskaya's result). The circle method gives
$$\frac{12}{\pi^2}\gamma P^3\log P+O(P^3),$$
where $$\gamma=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\left|
\int_{0}^{1}e^{2\pi i(z_1x^2+z_2x)}dx\right|^6dz_1dz_2$$
UPD: This arguments are valid for symmetrical system of equations
$$x^2+y^2+z^2=u^2+v^2+w^2,\qquad x+y+z=u+v+w.$$
The original system will take this form after changing variables $z\to -z$, $v\to -v$.
A: This may not give all the solutions, but it does give a 3-parameter family. Choose positive integers $a,b,c,d$ such that $$a+b=c+d,\quad a+c<b<a+d,\quad2a<b$$ Then $$b^2+(b-a-c)^2+(a+d-b)^2=c^2+d^2+(b-2a)^2$$ and $$b+(b-2a)+(a+d-b)=c+d+(b-a-c)$$
A: 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:
$$x=2s^2+(2q+2t+k)s+kt+qk+2qt$$
$$y=s^2+(q+k)s+qk+kt-t^2$$
$$z=s^2+(t+k)s+qk+kt-q^2$$
$$u=s^2+(q+t+k)s+qt+qk+kt$$
$$w=2s^2+(2q+2t+k)s+qt+qk+kt$$
$$v=s^2+ks+qk+kt-q^2-t^2$$
