Which functions are spherical derivatives?

Let us define the differential operator
$$
Z=x_1 partial_{x_2} – x_2 partial_{x_1}, $$
where $(x_1, x_2, x_3)$ are the standard Cartesian coordinates on $mathbb R^3$. I would like to characterize the functions $hin C^infty(mathbb S^{2})$ such that
$$tag{1}
h=Zf, qquad text{for some }fin C^infty(mathbb S^{2}).$$

The operator $Z$ is one of the generators of the rotation group $SO(3)$, in the sense that
$$tag{2}
Zf(x)=lim_{epsilonto 0}frac1epsilon big( f(R_epsilon x)-f(x)big), $$
where $R_epsilon$ is the matrix
$$
R_epsilon=begin{bmatrix} 
cos epsilon & -sin epsilon & 0 \
sin epsilon & cos epsilon & 0 \
0 & 0 & 1
end{bmatrix}.$$
The formula (2) implies that
$$int_{mathbb S^{2}} Zf(x), dS(x)=0, $$
where $dS$ denotes the standard Lebesgue measure on the sphere. Thus, a necessary condition for (1) to hold is that
$$tag{3}
int_{mathbb S^{2}} h, dS = 0.$$
Also, since $Z$ vanishes at $(0,0, pm 1)$, another necessary condition is
$$
tag{4} h(0,0,pm 1)=0.$$
Are these two last conditions also sufficient?