  This article can be viewed as a continuation of the articles arXiv:0912.3486
and arXiv:1012.3714 where the decomposable Lie algebras admitting half-flat
SU(3)-structures are classified. The new main result is the classification of
the indecomposable six-dimensional Lie algebras with five-dimensional
nilradical which admit a half-flat SU(3)-structure. As an important step of the
proof, a considerable refinement of the classification of six-dimensional Lie
algebras with five-dimensional non-Abelian nilradical is established.
Additionally, it is proved that all non-solvable six-dimensional Lie algebras
admit half-flat SU(3)-structures.
