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Let 
$M$ be an analytic connected 2-manifold with empty boundary, over the ground field 
$\mathbb{F}=\mathbb{R}$ or 
$\mathbb{C}$. Let 
$Y$ and 
$X$ denote differentiable vector fields on 
$M$. We say that 
$Y$ tracks 
$X$ if 
$[Y,X]=fX$ for some continuous function 
$f:\,M\rightarrow \mathbb{F}$. A subset 
$K$ of the zero set 
$\mathsf{Z}(X)$ is an essential block for 
$X$ if it is non-empty, compact and open in 
$\mathsf{Z}(X)$, and the Poincaré–Hopf index 
$\mathsf{i}_{K}(X)$ is non-zero. Let 
${\mathcal{G}}$ be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field 
$X$. Let 
$K\subset \mathsf{Z}(X)$ be an essential block. Assume that if 
$M$ is complex and 
$\mathsf{i}_{K}(X)$ is a positive even integer, no quotient of 
${\mathcal{G}}$ is isomorphic to 
$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then 
${\mathcal{G}}$ has a zero in 
$K$ (main result). As a consequence, if 
$X$ and 
$Y$ are analytic, 
$X$ is non-trivial, and 
$Y$ tracks 
$X$, then every essential component of 
$\mathsf{Z}(X)$ meets 
$\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.
