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 $abcdefg\neq 0$ ?
I do not think there is a known way of describing all of them, but here you'll find some further reading: http://mathworld.wolfram.com/CubicNumber.html
Regarding equation,  (a^3 + b^3 + c^3 + d^3 + e^3 + f^3 = g^3 )

The above equation has parametrization given below:

(a,b,c,d,e,f)^3=

[(6k^2+12k-18),(k^2-2k+49),(5k^2+38k+5),(8k^2+16k-24),
       (7k^2-14k-41),(10k^2+20k-30)]^3 = [(13k^2+22k+13)]^3= (g)^3

For k=2 we have:

(30,49,101,40,-41,50)^3=(109)^3