If x=y=z = t then (2t)^3 = t

t(8*t^2 - 1) = 0

t = {0, √2/4, -√2/4}

Three solutions are:

(x,y,z) = {(0,0,0), (√2/4,√2/4,√2/4), (-√2/4,-√2/4,-√2/4)}

There may be other solutions, but I consider it feasible that these are the only solutions.

Reasoning:  Looking at any one of the equations, in 3D it looks similar to the 2 dimensional equation y=x^3 except the pattern is extended above and below "the paper", as if you took a flat plane then bent it into a y=x^3 shape when viewed on end.  Only rotated relative to the x,y,z axes.

Now intersect 2 orthogonal versions of that structure and you would get a line curving through 3 dimensions.

Intersect that with one more orthogonal structure and I think you would get a finite number of points.

Obviously there may be more than just these 3 points, I look forward to more solutions or a proof that these are the full solution set.