On solvable subvarieties of the variety generated by the Witt algebra

The conjecture that the commutator subalgebra of any solvable algebra lying in the variety generated by the Lie algebra of vector fields on the line is nilpotent is disproved in the case when the ground field has zero characteristic. The algebra constructed turns out to be useful for describing all solvable subvarieties of the variety generated by the Lie algebra of vector fields on the line (it may be regarded as a Witt algebra).
It is proved that any such subvariety either contains this algebra, or consists of algebras with nilpotent commutator subalgebras. An essential role in the proof is played by a result that is of independent interest: a solvable variety consists of algebras with nilpotent commutator subalgebras if and only if all its algebras with degree of nilpotency at most three have this property.
Bibliography: 14 titles.