The first equation has many solutions which are cubes; in fact the integer cube roots form the sequence:  4, 7, 10, ..., 3k+1.

But I can't find any cubes formed by the second (cubic) equation.

The explanation may be as simple as this:

The cubic equation has 1 real root and 2 complex roots; their sum is 0 and their product is -5/3; if this expression could be a perfect cube, the 3 roots would all be the same.