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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 $W_n(K)$ isthe 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 $P_n.$ Suchderivations define automorphisms of the ring $P_n$ and were studied by manyauthors. The subalgebra $u_n(K) $ is contained in another interestingsubalgebra $s_n(K)=(P_0+x_1P_0)\partial_1+\cdots+(P_{n-1}+x_nP_{n-1})\partial_n,$ which is solvable of the derived length $2n$that is the maximum derived length of solvable subalgebras of $W_n(K).$ It isproved that $u_n(K)$ is a maximal locally nilpotent subalgebra and $s_n(K)$ isa maximal solvable subalgebra of the Lie algebra $W_n(K).$

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