  This paper is devoted to the study of the global existence of smooth
solutions for the 3+1 dimensional Einstein-Klein-Gordon systems with a $U(1)
\times \mathbb{R}$ isometry group for a class of regular Cauchy data. In our
first paper \cite{chen}, we reduce the Einstein equations to a 2+1 dimensional
Einstein-wave-Klein-Gordon system. And we show that the first possible
singularity can only occur at the axis. In this paper, we give a proof for the
global regularity for the 2+1 dimensional system. Firstly, we show the
non-concentration of the energy near the first possible singularity. Then, we
prove that the global regularity holds for initial data with small energy.
