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New smooth solutions of the Strominger system with non vanishing flux,non-trivial instanton and non-constant dilaton based on the quaternionicHeisenberg group are constructed. We show that through appropriate contractionsthe solutions found in the $G_2$-heterotic case converge to the heteroticsolutions on 6-dimensional inner non-K\"ahler spaces previously found by theauthors and, moreover, to new heterotic solutions with non-constant dilaton indimension 5. All solutions satisfy the heterotic equations of motion up to thefirst order of $\alpha^{\prime}$.

The quaternionic Heisenberg group and Heterotic String Solutions with non-constant dilaton in dimensions 7 and 5