Abstract: The action of effective modified gravity with explicit nondynamical background field couplings is studied. In particular, we focus on the Ricci and scalar sectors of the gravitational SME. We perform the ADM formulation of such a modified gravity system and found new boundary terms to construct the perturbative Hamiltonian. We show that the constraint structure is altered and identify both the Hamiltonian and momentum constraints with the normal-normal and normal-space projections of the modified Einstein tensor along the foliation. We show the equivalence of the dynamics of the theory obtained in the Hamiltonian formulation using the Hamilton-Jacobi equations and the one contained in the spatial subspace of the Einstein equation. Our goal for this talk is to summarize and explain the progress achieved and comment on the unsolved key questions.