  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
$f\colon M\rightarrow\mathbb F$. Let $K$ be a compact subset of the zero set
${\mathsf Z}(X)$ such that ${\mathsf
  Z}(X)-K$ is closed, with nonzero Poincar\'e-Hopf index (for example
$K={\mathsf Z}(X)$ when $M$ is compact and $\chi(M)\neq 0$) and let $\mathcal
G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$.
\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 $\mathcal G$ has some zero in $K$.
\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.
