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International audienceIn 1964, 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, C. Bonatti proposed in 1992 a local version, replacing the Euler characteristic by the Poincaré–Hopf index of a vector field $X$ in a region $U$, denoted by $ \operatorname{Ind}(X,U)$; he asked: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 $Y$?A positive answer was given in the case where $X$ and $Y$ are real analytic, in the same article where the above question was posed.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 for analytic vector fields should hold in the $C^1$ setting

Existence of common zeros for commuting vector fields on three manifolds