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We prove the existence of a one-parameter family of nearly parallel$G_2$-structures on the manifold $S^3\times \mathbb R^4$, which are mutuallynon isomorphic and invariant under the cohomogeneity one action of the group$SU(2)^3$. This family connects the two locally homogeneous nearly parallel$G_2$-structures which are induced by the homogeneous ones on the sphere $S^7$.

Nearly parallel $G_2$-structures with large symmetry group