Lie algebras of derivations with large abelian ideals

Let $\mathbb K$ be a field of characteristic zero, $A=\mathbb{K}[x_{1},\dots,x_{n}]$ the polynomial ring and $R=\mathbb{K}(x_{1},\dots,x_{n})$ the field of rational functions. The Lie algebra ${\widetilde W}_{n}(\mathbb{K}):=\operatorname{Der}_{\mathbb{K}}R$ of all $\mathbb{K}$-derivation on $R$ is a vector space (of dimension n) over $R$ and every its subalgebra $L$ has rank $\operatorname{rk}_{R}L=\dim_{R}RL$. We study subalgebras $L$ of rank $m$ over $R$ of the Lie algebra $\widetilde{W}_{n}(\mathbb{K})$ with an abelian ideal $I\subset L$ of the same rank $m$ over $R$. Let $F$ be the field of constants of $L$ in $R$. It is proved that there exist a basis $D_1,\dots,D_m$ of $FI$ over $F$, elements $a_1,\dots,a_k\in R$ such that $D_i(a_j)=\delta_{ij}$, $i=1,\dots,m$, $j=1,\dots,k$, and every element $D\in FL$ is of the form $D=\sum_{i=1}^{m}f_i(a_1,\dots,a_k)D_i$ for some $f_i\in F[t_1,\dots,t_k]$, $\deg f_i\leq 1$. As a consequence it is proved that $L$ is isomorphic to a subalgebra (of a very special type) of the general affine Lie algebra $\mathrm{aff}_{m}(F)$.