  Necessary and sufficient conditions to the existence of a hermitian
connection with totally skew-symmetric torsion and holonomy contained in SU(3)
are given. Non-compact solution to the supergravity-type I equations of motion
with non-zero flux and non-constant dilaton is found in dimensions 6.
Non-conformally flat non-compact solutions to the supergravity-type I equations
of motion with non-zero flux and non-constant dilaton are found in dimensions 7
and 8. A Riemannian metric with holonomy contained in $G_2$ arises from our
considerations and Hitchin's flow equations, which seems to be new. Compact
examples of $SU(3), G_2$ and $Spin(7)$ instanton satisfying the anomaly
cancellation conditions are presented.
