  We study the fillability (or embeddability) of $CR$ structures under the
gauge-fixed Cartan flow. We prove that if the initial $CR$ structure is
fillable with nowhere vanishing Tanaka-Webster curvature and free torsion, then
it keeps having the same property after a short time. In the Appendix, we show
the uniqueness of the solution to the gauge-fixed Cartan flow.
