  Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of
characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$
be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all
right operator identities of $\mathscr{L}_n$ and prove that the set of all
algebras $\mathscr{L}_n$, where $n\geq 1$, generates the variety of all
left-symmetric algebras. We also describe a class of general (not only right
operator) identities for $\mathscr{L}_n$.
