Something of a mystery surrounds the French LIP Company, their precision milling machine being built under a licence from "Carpo", and manufactured by LIP Industrie at 7 rue des Chalets, Bresancon, Issoudun, Grenoble in France. LIP grinders, made by Lipemec Machine Tools, were made at 13 rue de Contrevaux, 25290 Orans to the south east of Besancon, in the French Jura. So, which Company made the LIP lathe is not known, and it might have been yet another in the group. It is known that the milling machine was built under licence from Carpo (originally Louis Carpano and then Carpano & Pons, Industries Clauses, France), an organisation that held various interesting patents including ones associated with the first electric razors, a ski binding, cigarette lighters, car electric windscreen wipers, millers, grinding machines and, probably most famous of all, Mitchell fishing reels. LIP was closely connected to the horological industry and many of their products (they were known for high-precision grinding machines) were employed in watch and clock production processes.
Looking to be of the usual very high standard of other machines tools by LIP, the ways were hand scraped and the headstock spindle able to take 8 mm collets of the watchmaker's type, -but much longer and with an extended length of thread. The tailstock also took collets, but tiny ones with a 4 or 5 mm body diameter of an unusual type.
The bearing portion of the tailstock was spring loaded - but on the sole known surviving example, appears to have nothing adjust the travel.
Not intended for general watchmaking work, the LIP  would have been employed for some special purpose in a  production process. Running in air bearings, the spindle was capable of running at very high speeds for extended periods - so presumably ideal for polishing pivots and similar operations.
If you can throw any light on the function of this lathe, the writer would be pleased to
hear from you.