Show $S^2$ cannot have a smooth vector field with two zeros that are either both sources or both sinks Backround
I have just learned the Poincaré-Hopf Index Theorem which says that if $\overrightarrow{v}$ is a smooth vector field on a compact,
oriented manifold $X$ with only finitely many zeros, then the global sum of the
indices of $\overrightarrow{v}$ equals the Euler characteristic of $X$. This is great, but it may not be the whole story regarding the "rules" of which combinations of different qualitative types of zeros a vector field can and can't have on a given space. 
Particular Question
Edit: my "particular question" assumed something that was wrong, so only the general question makes sense
For example, my intuition tells me that on $S^2$ one cannot have just two zeros where both are sources or both are sinks. But this is not ruled out by Poincaré-Hopf since index doesn't distinguish between source and sink in two dimensions. So how can we rigorously rule this out? 
General Quesiton
What are the key theorems or theories used for which combinations of different qualitative types of zeros a vector field can and can't have on a given space/manifold? I am generally interested in simple spaces such as balls and spheres (in arbitrary dimensions) and also cartesian products thereof. I am not so interested in spaces with complicated combinations of holes of various dimensions and so on. So far I have been reading Guilleman and Pollack's Differential Topology. 

Consider the following vector field $\vec v$ on $S^2$. Using standard spherical coordinates $\theta$, $\phi$, and let $\hat u_\theta$ and $\hat u_\phi$ be the unit vectors in the $\theta$ and $\phi$ directions, $\vec v$ is given by:
$$ \vec v =
\begin{cases}
v_0\sin(2\theta)\hat u_\theta +v_0\sin(\theta)\hat u_\phi &\text{if $\theta\ne0$},\\
0 & \text{if $\theta=0$},
\end{cases}$$
where $v_0$ is a positive constant. This vector field has only two zeros (at the N and S poles) and both zeros are sources (if you make $v_0<0$ then they are both sinks).
