  Let \Sigma be a compact oriented surface immersed in a four dimensional
K\"ahler-Einstein manifold M. We consider the evolution of \Sigma in the
direction of its mean curvature vector. It is proved that being symplectic is
preserved along the flow and the flow does not develop type I singularity. When
M has two parallel K\"ahler forms \omega' and \omega'' that determine different
orientations and\Sigma is symplectic with respect to both \omega' and \omega'',
we prove the mean curvature flow of \Sigma exists smoothly for all time. In the
positive curvature case, the flow indeed converges at infinity.
