  Unless another thing is stated one works in the $C^\infty$ 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
$f\colon M\rightarrow\mathbb R$. A subset $K$ of the zero set ${\mathsf Z}(X)$
is an essential block for $X$ if it is non-empty, compact, open in ${\mathsf
Z}(X)$ and its Poincar\'e-Hopf index does not vanishes. One says that $X$ is
non-flat at $p$ if its $\infty$-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 $X$.
