  We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with
Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R +
S^2_0(R^7) + R^7 in the sense of Fern\'andez and Gray. This relies on the
construction of some G_2 solvmanifolds, whose Levi-Civita connection is known
to give a parallel spinor, admitting a 2-parameter family of metric connections
with non-zero skew-symmetric torsion that has parallel spinors as well. The
family turns out to be a deformation of the Levi-Civita connection. This is in
contrast with the case of compact scalar-flat Riemannian spin manifolds, where
any metric connection with closed torsion admitting parallel spinors has to be
torsion-free.
