  We address the following conjecture about the existence of common zeros for
commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting
vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such
that $X$ does not vanish on the boundary of $U$ and has a non vanishing
Poincar\'e-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$.
We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic
orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We
also prove the conjecture, still in the $C^3$ setting, assuming that the flow
$Y$ leaves invariant a transverse plane field. These results shed new light on
the $C^3$ case of the conjecture.
