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 $fcolon Mrightarrowmathbb F$. Let $K$ be a compact subset of the zero set ${mathsf Z}(X)$ such that ${mathsf Z}(X)-K$ is closed, with nonzero Poincare-Hopf index (for example $K={mathsf Z}(X)$ when $M$ is compact and $chi(M) eq 0$) 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.
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