We classify left-invariant half-flat SU(3)-structures on $S^3\times S^3$, using the representation theory of SO(4) and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy $\mathrm{G}_2$ obtained on 7-manifolds with equidistant $S^3\times S^3$ hypersurfaces. The generic case is analysed numerically.

Keywords: $\mathrm{G}_2$- and SU(3)-structures, Einstein and Ricci-flat manifolds, special and exceptional holonomy, stable forms, superpotential.