A family of maximal subalgebras of the Lie algebra~$W_n(K)$

Let $K$ be an algebraically closed field of characteristic zero and $P_n=K[x_1,\dots ,x_n]$ the polynomial ring. Any $K$-derivation $D$ on $P_n$ is of the form $D={\sum }_{i=1}^n{f}_i(x_1,\dots ,x_n)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i},$ where $f_i\in P_n\mathrm{.}$ All such derivations form the Lie algebra $W_n(K)$ over the field $K$. We prove that for $s=1,\dots ,n-1$ the subalgebra $m_s(K)=\{{\sum }_{i=1}^s{f}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}+{\sum }_{j=s+1}^n{g}_j\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\mid f_i\in P_s,g_j\in P_n\}$ is a maximal subalgebra of~$W_n(K)$. The ideal $I_s=\left\{{\sum }_{j=s+1}^n{g}_j\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\right\}$ of $m_s(K)$ is isomorphic to the Lie algebra $P_s\otimes \mathrm{D}\mathrm{e}\mathrm{r}(K[x_{s+1},\dots ,x_n])$ and $m_s(K)\mathrm{/}{I}_s\simeq W_s(K)$. The Lie algebra $W_n(K)$ is also the free module over the ring $P_n\mathrm{.}$ Therefore, for any set $S\subseteq W_n(K)$ the rank $rk(S)$ (over $P_n$) is defined. Some properties of maximal subalgebras of rank $n$ in $W_n(K)$ are pointed out.