Maximal subalgebras of the Lie algebra $W_n(\mathbb{K})$

Let $K$ be an algebraically closed field of characteristic zero, $A=K[x_1,\dots ,x_n]$ the polynomial ring in $n$ variables, and let $W_n(K)$ be the Lie algebra of all $K$-derivations of $A\mathrm{.}$ This Lie algebra also is the free $A$-module of rank $n$ over the ring $A,$ so every subalgebra of $W_n(K)$ has a rank $\le n$ over $A\mathrm{.}$ We prove that every maximal subalgebra of rank $\le n$ of $W_n(K)$ is a simple Lie algebra. If a maximal subalgebra $L\subset W_n(K)$ has rank $n$ and is a submodule of $W_n(K)$ then $L$ is not simple. Moreover, $L$ is of the form $L=\{D\in W_n(K)\mathrm{\mid }D(I)\subseteq I\}$ for some ideal $I$ of the ring $A\mathrm{.}$ It is also proved that, for a simple derivation $D$ on the ring $K[x,y]$, the subalgebra $K[x,y]D$ is a maximal subalgebra of $W_2(K)\mathrm{.}$