The generality of closed $G_2$ solitons

The local generality of the space of solitons for the Laplacian flow of closed $G_2$-structures is analyzed, and it is shown that the germs of such structures depend, up to diffeomorphism, on 16 functions of 6 variables (in the sense of E. Cartan). The method is to construct a natural exterior differential system whose integral manifolds describe such solitons and to show that it is involutive in Cartan’s sense, so that Cartan-Kahler theory can be applied. Meanwhile, it turns out that, for the more special case of gradient solitons, the natural exterior differential system is not involutive, and the generality of these structures remains a mystery.