  We present the necessary and sufficient conditions of the well-posedness of
the initial value problem for certain fourth-order linear dispersive systems on
the one-dimensional torus. This system is related with a dispersive flow for
closed curves into compact Riemann surfaces. For this reason, we study not only
the general case but also the corresponding special case in detail. We apply
our results on the linear systems to the fourth-order dispersive flows. We see
that if the sectional curvature of the target Riemann surface is constant, then
the equation of the dispersive flow satisfies our conditions of the
well-posedness.
