  We show that free algebras of the variety of algebras generated by the Witt
algebra $W_n$, the left-symmetric Witt algebra $L_n$, and the symplectic
Poisson algebra $P_n$ can be described as subalgebras of differential
polynomial algebras with respect to appropriately defined products. Using these
representations, we prove that $W_n$, $L_n$, $P_n$, and the free algebras of
the varieties of algebras generated by these algebras are equationally
Noetherian.
