  In $64$ E. Lima proved that commuting vector fields on surfaces with non-zero
Euler characteristic have common zeros. Such statement is empty in dimension
$3$, since all the Euler characteristics vanish. Nevertheless,
\cite{Bonatti_analiticos} proposed a local version, replacing the Euler
characteristic by the Poincar\'e-Hopf index of a vector field $X$ in a region
$U$, denoted by $\operatorname{Ind}(X,U)$; he asked:
  \emph{Given commuting vector fields $X,Y$ and a region $U$ where
$\operatorname{Ind}(X,U)\neq 0$, does $U$ contain a common zero of $X$ and
$Y$?}
  \cite{Bonatti_analiticos} gave a positive answer in the case where $X$ and
$Y$ are real analytic.
  In this paper, we prove the existence of common zeros for commuting $C^1$
vector fields $X$, $Y$ on a $3$-manifold, in any region $U$ such that
$\operatorname{Ind}(X,U)\neq 0$, assuming that the set of collinearity of $X$
and $Y$ is contained in a smooth surface. This is a strong indication that the
results in \cite{Bonatti_analiticos} should hold for $C^1$-vector fields.
