Existence of common zeros for commuting vector fields on $3$-manifolds II. Solving global difficulties

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