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
$$a_1+a_2+a_3=0,\qquad a_1b_1+a_2b_2+a_3b_3=0.$$