We study supersymmetric solutions in 7- and 8-dimensional Abelian heterotic supergravity theories. In dimension 7, the solutions are described by  with torsion equations. When a  manifold has principal orbits  the equations are reduced to ordinary differential equations with four radial functions. For these equations, we obtain explicit ALC metrics with -bolt and -bolt singularities. In dimension 8, we study supersymmetric solutions to  with torsion equations associated with 3-Sasakian manifolds by using a similar method to the case

   supergravity.