  Curvature properties of the characteristic connection on an integrable $G_2$
space are investigated. It is proved that an integrable $G_2$ manifold has
closed torsion if and only if the Ricci tensor of the characteristic connection
is equal to the covariant derivative of the Lee form and in this case the
integrable $G_2$ structure is of constant type. It is shown, that a compact
integrable $G_2$ manifold with closed torsion is Ricci flat if and only if
either the norm of the torsion is constant or the Riemannian scalar curvature
is constant. It is observed that any compact integrable $G_2$ manifold with
closed torsion 3-form is a generalized gradient Ricci soliton and this is
equivalent to a certain vector field to be parallel with respect to the torsion
connection. In particular, this vector field is an infinitesimal authomorphism
of the $G_2$ strucure. Compact examples with bi-$G_2$ structures or generalized
integrable $G_2$ manifolds are given.
  It is shown that on an integrable $G_2$ space of constant type the curvature
of the characteristic connection $R\in S^2\Lambda^2$ with vanishing Ricci
tensor if and only if the three-form torsion is parallel with respect to the
Levi-Civita and to the characteristic connection simultaneously. In particular,
the conditions $R\in S^2\Lambda^2, Ric=0$ are equivalent to the condition that
the curvature of the characteristic connection satisfies the Riemannian first
Bianchi identity. In this case the torsion 3-form is harmonic.
