Answer
None.
Work Step by Step
Step 1. To test symmetry with respect to the x-axis, replace $(x,y)$ with $(x,-y)$, we have: $(-y)^3=x+4$ which is different from the original, thus it is not symmetric with respect to the x-axis.
Step 2. To test symmetry with respect to the y-axis, replace $(x,y)$ with $(-x,y)$, we have: $(y)^3=-x+4$ which is different from the original, thus it is symmetric not with respect to the y-axis.
Step 3. To test symmetry with respect to the origin, replace $(x,y)$ with $(-x,-y)$, we have: $(-y)^3=-x+4$ which is different from the original, thus it is not symmetric with respect to the origin.
Step 4. Thus the equation has no symmetry with respect to the x-axis, the y-axis, or the origin.