  Let $(M,g)$ be an asymptotically flat Riemannian $3$-manifold. We provide a
short new proof based on Lyapunov-Schmidt reduction of the existence of an
asymptotic foliation of $(M,g)$ by constant mean curvature spheres. In the case
where the scalar curvature of $(M,g)$ is non-negative, we prove that the leaves
of this foliation are the only large stable constant mean curvature spheres
that enclose the center of $(M,g)$. This had been shown previously under more
restrictive assumptions and using a different method by S. Ma. We also include
a new proof of the fact that the geometric center of mass of the foliation
agrees with the Hamiltonian center of mass of $(M,g)$.
