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

The Riemannian curvature identities for the torsion connection on $Spin(7)$-manifold and generalized Ricci solitons