On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)

Let $K$ be an algebraically closed field of characteristic zero, $P_n=K[x_1, ..., x_n]$ the polynomial ring, and $W_n(K)$ the Lie algebra of all $K$-derivations on $P_n$. One of the most important subalgebras of $W_n(K)$ is the triangular subalgebra $u_n(K) = P_0\partial_1+\cdots+P_{n-1}\partial_n$, where $\partial_i:=\partial/\partial x_i$ are partial derivatives on $P_n$. This subalgebra consists of locally nilpotent derivations on $P_n.$ Such derivations define automorphisms of the ring $P_n$ and were studied by many authors. The subalgebra $u_n(K) $ is contained in another interesting subalgebra $s_n(K)=(P_0+x_1P_0)\partial_1+\cdots +(P_{n-1}+x_nP_{n-1})\partial_n,$ which is solvable of the derived length $2n$ that is the maximum derived length of solvable subalgebras of $W_n(K).$ It is proved that $u_n(K)$ is a maximal locally nilpotent subalgebra and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K).$