Coclosed $$G_2$$-structures on $$\text {SU}(2)^2$$-invariant cohomogeneity one manifolds

We consider two different $$\text {SU}(2)^2$$ SU ( 2 ) 2 -invariant cohomogeneity one manifolds, one non-compact $$M=\mathbb {R}^4 \times S^3$$ M = R 4 × S 3 and one compact $$M=S^4 \times S^3$$ M = S 4 × S 3 , and study the existence of coclosed $$\text {SU}(2)^2$$ SU ( 2 ) 2 -invariant $$G_2$$ G 2 -structures constructed from half-flat $$\text {SU}(3)$$ SU ( 3 ) -structures. For $$\mathbb {R}^4 \times S^3$$ R 4 × S 3 , we prove the existence of a family of coclosed (but not necessarily torsion-free) $$G_2$$ G 2 -structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed $$G_2$$ G 2 -structure constructed from a half-flat $$\text {SU}(3)$$ SU ( 3 ) -structure is in this family. For $$S^4 \times S^3$$ S 4 × S 3 , we prove that there are no $$\text {SU}(2)^2$$ SU ( 2 ) 2 -invariant coclosed $$G_2$$ G 2 -structures constructed from half-flat $$\text {SU}(3)$$ SU ( 3 ) -structures.