  Let $\mathbb F$ be a field $\mathbb F $ of characteristic zero. Let
$W_{n}(\mathbb F)$ be the Lie algebra of all $\mathbb F$-derivations with the
Lie bracket $[D_1, D_2]:=D_1D_2-D_2D_1$ on the polynomial ring $\mathbb F [x_1,
\ldots , x_n]$. The problem of classifying finite dimensional subalgebras of
$W_{n}(\mathbb F)$ was solved if $ n\leq 2$ and $\mathbb F=\mathbb C$ or
$\mathbb F=\mathbb R.$ We prove that this problem is wild if $n\geq 4$, which
means that it contains the classical unsolved problem of classifying matrix
pairs up to similarity. The structure of finite dimensional subalgebras of
$W_{n}(\mathbb F)$ is interesting since each derivation in case $\mathbb
F=\mathbb R$ can be considered as a vector field with polynomial coefficients
on the manifold $\mathbb R^{n}.$
