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Let  $\sum $ be a compact oriented surface immersed in a four dimensional K¨ahler-Einsteinmanifold $(M, ω)$. We consider the evolution of $\sum$ 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  $\sum$ is symplectic with respect to both $\omega " $ and $\omega ''$, we prove the mean curvature flow of $\sum$  exists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

Mean curvature flow, Einstein four-manifolds,