Before revision:  Key words. time-inconsistency, stochastic linear-quadratic control, uniqueness of equilibrium control, ] forward--backward stochastic differential equation, mean--variance portfolio selection  .
After revision:  Key words. time-inconsistency, stochastic linear-quadratic control, uniqueness of equilibrium control, ]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equation, mean--variance portfolio selection .