Lipid nanotube-vesicle networks are important channels for intercellular communication and transport of matter. Experimentally observed in neighboring mammalian cells, but also reproduced in model membrane systems, a broad consensus exists on their formation and stability. Lipid membranes must be composed of at least two components, each stabilizing low (generally a phospholipid) and high curvatures. Strong anisotropy or enhanced conical shape of the second amphiphilic component is crucial for the formation of nanotunnels. Anisotropic driving forces generally favor nanotube protrusions from vesicles. In the present work, we report the unique case of topologically-connected nanotubes-vesicles obtained in the absence of directional forces, in single-component membranes, composed of an anisotropic bolaform glucolipid, above its melting temperature, Tm. Cryo-TEM and fluorescence confocal microscopy show the interconnection between vesicles and nanotubes in a single-phase region, between 60° and 90°C under diluted conditions. Solid-state NMR, necessarily recorded on the dry powder, yet demonstrates that the glucolipid can simultaneously assume two distinct configurations, head-head and head-tail. These arrangements, seemingly of comparable energy above the Tm, could explain the existence and stability of the topologically-connected vesicles and nanotubes, which are generally not observed for classical single-component phospholipid-based membranes above their Tm.