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, ] forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in \mbox{%DIFAUXCMD HJZ mean--variance portfolio selection .