HA9, Problem6

[Shentao YANG]

Can any explain in detail the reason behind the hint "solution must be a harmonic polynomial of degree $3$ and it should depend only on $x^2+y^2+z^2$ and $z$"?
Concretely, there are $3$ question need to be addressed:
$(1)$ Why the solution need to be a harmonic polynomial?
$(2)$ Why it should have degree $3$?
$(3)$ Why it should depend only on $x^2+y^2+z^2$ and $z$ but not the other terms?
The solution from the previous year basically made no reasonable explanation and just copied words from the textbook

http://forum.math.toronto.edu/index.php?topic=731.0

[Victor Ivrii]

Consider it as a Hint. We have not finished harmonic polynomials on the lecture.
Also, since problem depends only on $x^2+y^2$ and $z$ (is invariant with respect to rotations around $z$ axis), so does the solution