Question: How to choose the integer number a, b, c, d, e, f, k to the equation has three different integer solutions?

The equation 

solve(surd(-2*x+4,3)+surd(x+2,3)+surd(x -6> ,3)=0);

has three different integer solutions.

I want to choose the integer number a, b, c, d, e, f to the equation 

(a*x + b)^(1/3) + (c*x + d)^(1/3) + (e*x + f)^(1/3) =0 

has three different integer solutions.

I tried

restart:

L:=[]:

for a from -10 to -1  do

for b from -10 to 20  do

for c from 1 to 10  do

for d from 1 to 10  do

for e from 1 to 10  do

for f from -10 to 10  do

if  -b*c + a*d - b*e + a*f = 0 and -a*d + b*c - e*d + c*f = 0 and - a*f + b*e - c*f + d*e = 0  and   type(a,integer) and type(b,integer) and type(c,integer) and type(d,integer) and type(e,integer) and type(f,integer) then

L:=[op(L), [a,b,c,d,e,f]] fi;

od: od: od:od: od: od:

nops(L); 

L;

 This is a necessary condition, but not sufficient condition. How to get the sufficient condition?

Open problem. How to choose the integer number a, b, c, d, e, f,  k to the equation 

(a*x + b)^(1/3) + (c*x + d)^(1/3) + (e*x + f)^(1/3) = g

has three different integer solutions?