Brillinger  gives necessary and sufficient conditions for a model to be invariant under a Lie group of transformations. The problems that can be handled by his conditions are surveyed, and found effectively to be restricted to one-dimensional problems amendable to Lindley's  method and to problems connected with conflicts between Bayes' and fiducial theory. The problem of finding the general model invariant under a given group is proposed. Brillinger's theorem produces differential equations for the model. A general solution can be obtained by direct methods.
"Statistical Models and Invariance." Ann. Math. Statist. 38 (4) 1061 - 1067, August, 1967. https://doi.org/10.1214/aoms/1177698775