The estimation error covariance of a fixed-point smoother in the steady-state, which has the steady-state solution of the Kalman filter as the initial condition, is considered for both continuous- and discrete-time systems. Applying some results on a stabilizing solution for a forward-pass fixed-interval smoother, a necessary and sufficient condition is given for assuring the existence of a unique stabilizing solution for such a fixed-point smoother. It is then shown that the resulting condition is equivalent to a well-known condition for the existence of a unique stabilizing solution of the Kalman filter or of the backward information filter.