LAUSR

Unless another thing is stated one works in the $$C^\infty $$ C ∞ category and manifolds have empty boundary. Let X and Y be vector fields on a manifold M . We say that Y tracks X if $$[Y,X]=fX$$ [ Y , X ] = f X for some continuous function $$f:M\rightarrow \mathbb {R}$$ f : M → R . A subset K of the zero set $${\mathsf {Z}} (X)$$ Z ( X ) is an essential block for X if it is non-empty, compact, open in $${\mathsf {Z}}(X)$$ Z ( X ) and its Poincaré-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 $${\mathsf {Z}}(X)$$ 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 .