Primary singularities of vector fields on surfaces

Unless another thing is stated one works in the $C^{\mathrm{\infty }}$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\text{\hspace{0.17em}⁣}:M\to \mathbb{R}$. A subset $K$ of the zero set $\mathsf{Z}(X)$ is an essential block for $X$ if it is non-empty, compact, open in ${\mathsf Z}(X)$anditsPoincar\acute{e}-Hopfindexdoesnotvanishes\mathrm{.}Onesaysthat$X$ is non-flat at $p$ if its $\mathrm{\infty }$-jet at $p$ is non-trivial. A point $p$ of $\mathsf{Z}(X)$ is called a primary singularity of $X$ if any vector field defined about $p$ and tracking $X$ vanishes at $p$. This is our main result: Consider an essential block $K$ of a vector field $X$ defined on a surface $M$. Assume that $X$ is non-flat at every point of $K$. Then $K$ contains a primary singularity of $X$. As a consequence, if $M$ is a compact surface with non-zero characteristic and $X$ is nowhere flat, then there exists a primary singularity of $X$.