A. The sheaf of solutions (J ,∇) of the Hessian equation on a gauge structure (M, ∇) is a key ingredient for understanding important properties from the cohomological point of view. In this work, a canonical representation of the group associated by Lie third's theorem to the Lie algebra formed by the sections of (J, ∇) is introduced. On the foliation it defines, a characterization of compact hyperbolic leaves is then obtained. We conclude that these leaves are equipped with a statistical model structure.