Let $V$ be a vector field on $S^3$ such that its singularity points, namely the points at which the vector field vanishes, are only *sinks* or *sources* (i.e. the field is converging or diverging). Is there an upper bound on the number $N$ of sinks and sources? Is this number 2? Is it possible that N=1?

There are examples of non-singular fields (Hopf fibering) for which N=0 and simple examples with N=2 (gradient of height function). I am asking if there are other possibilities.

The analogous question on $S^2$ has negative answer by the Poincaré-Hopf theorem which states that N=2.

Here is my attempt: If there is an integral curve of the vector field which joins a source A with a sink B then every integral curve from A must end in B (intuitively, this follows from an argument on connectedness of the boundary of a small ball centered at A) thus in this case N=2, and we are left with the possibility that all the sinks and sources are 'separated' by invariant sets, but is this really possible?