Central extension of Virasoro type subalgebras of the Zamolodchikov--$w_{\infty}$ Lie algebra

It is known that the centerless Zamolodchikov--$w_{\infty}$ $*$--Lie algebra of conformal field theory does not admit nontrivial central extensions, but the Witt $*$--Lie algebra, which is a sub--algebra of $w_{\infty}$, admits a nontrivial central extension: the Virasoro algebra. Therefore the following question naturally arises: {\it are there other natural sub--algebras of $w_{\infty}$ which admit nontrivial central extensions other than the Virasoro one?} We show that for certain infinite dimensional closed subalgebras of $w_{\infty}$, which are natural generalizations of the Witt algebra the answer is negative.