Zero sets of Lie algebras of analytic vector fields on real and complex 2-dimensional manifolds, II

On a real ($\mathbb{F}=\mathbb{R}$) or complex ($\mathbb{F}=\mathbb{C}$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function $f\text{\hspace{0.17em}⁣}:M\to \mathbb{F}$. Let $K$ be a compact subset of the zero set $\mathsf{Z}(X)$ such that ${\mathsf Z}(X)-K$isclosed,withnonzeroPoincar\acute{e}-Hopfindex(forexample$K={\mathsf Z}(X)$when$M$iscompactand$\chi(M)\neq 0$)andlet$\mathcal G$beafinite-dimensionalLiealgebraofanalyticvectorfieldson$M$. \smallskip {\bf Theorem.} Let $X$ be analytic and nontrivial. If every element of $\mathcal{G}$ tracks $X$ and, in the complex case when ${\mathsf{i}}_K(X)$ is positive and even no quotient of $\mathcal{G}$ is isomorphic to ${\mathfrak {s}}{\mathfrak {l}} (2,\mathbb C)$,then$\mathcal G$hassomezeroin$K$. \smallskip {\bf Corollary.} If $Y$ tracks a nontrivial vector field $X$, both of them analytic, then $Y$ vanishes somewhere in $K$. \smallskip Besides fixed point theorems for certain types of transformation groups are proved. Several illustrative examples are given.