The bilayers of some typical biological membrane lipids such as PC and DGDG disintegrate in a large excess of water to form an optically invisible dispersive bilayer phase. `Dark bodies' can be reversibly precipitated from it by raising the temperature. The dispersive phase probably consists of `knotted sticks', i.e. very thin nodular tubes of bilayer. After reviewing pertinent experimental and theoretical work we report on the discovery of a lower consolute point near room temperature in DGDG/water systems. Its existence shows that the dispersive phase and the dark bodies belong to the same fragmented (or nodular) bilayer state, representing its expanded and condensed phases, respectively, above the critical temperature.