This is a pincushion. Find the intercepts first.
When x=0: y⅔=4⅔. Cube each side: y2=42, so y=±4. Intercepts at (0,-4) and (0,4).
When y=0, similarly, x=±4, intercepts at (-4,0) and (4,0). These 4 points are vertices. The figure is also symmetrical.
[x⅔=4⅔-y⅔=(4⅓+y⅓)(4⅓-y⅓)=(∛4+∛y)(∛4-∛y).]
When y=1, we get x⅔=4⅔-1=1.52 approx. x2=1.523=3.51, x=±√3.51=±1.87 approx.
Also, when x=1, y=±1.87. This gives you 4 more points to plot: (1,1.87), (1,-1.87), (1.87,1), (-1.87,1).
And, because of symmetry, you can also plot (-1.1.87), (-1,-1.87), (1.87,-1), (-1.87,-1).
From the 12 points you can sketch the graph.
Now you should be able to visualise the results of rotation to generate a solid.