Describe the nonzero integer solutions to the equation $a^3 + b^3 + c^3 + d^3 + e^3 + f^3 + g^3 =0$ Can someone describe all the integer solutions to the above equation such that $abcdefg\neq 0$ ?

For the equation.
$$x_1^3+x_2^3+x_3^3+x_4^3+x_5^3+x_6^3=x_7^3$$
You can write a fairly simple formula.
$$x_1=t^2-3(k+s)(p+t)-3p^2+2u$$
$$x_3=t^2-3(p+s)(k+t)-3k^2+2u$$
$$x_5=t^2-3(p+k)(s+t)-3s^2+2u$$
$$x_2=2t^2+3((k+s)(p-t)-2pt)+3p^2+u$$
$$x_4=2t^2+3((p+s)(k-t)-2kt)+3k^2+u$$
$$x_6=2t^2+3((p+k)(s-t)-2st)+3s^2+u$$
$$x_7=3(t^2-2(p+k+s)t+u)$$
where, 
$$u=3(p^2+k^2+s^2)$$
Cube certainly look nice, but I prefer to solve such equations. Look cumbersome, but the solution much simpler.  The sum of the cubes and the amount of combinations.
