  Conformally compact and complete smooth solutions to the Strominger system
with non vanishing flux, non-trivial instanton and non-constant dilaton using
the first Pontrjagin form of the (-)-connection} on 6-dimensional non-Kaehler
nilmanifold are presented. In the conformally compact case the dilaton is
determined by the real slices of the elliptic Weierstrass function. The dilaton
of non-compact complete solutions is given by the fundamental solution of the
Laplacian on $R^4$.
