Resident in France, this lathe appears to be relatively common with three being offered for sale during 2023. With a centre height of perhaps 100 mm and admitting approximately 350 mm between centres, the design of the lathe was certainly unusual -  the headstock and tailstock were offset towards the rear of the bed a system of changewheel drive to the leadscrew that must, surely, remain unique.
One can understand the designer's desire to save money by using a large, 3-step round-belt driven drive pulley overhung on the end of the headstock spindle - but its use then interfered with fitting a normal train of gears running down the outside face of the headstock. While the English EXE lathe overcame this problem, the designer of our "unknown" used a different approach with that part of the headstock spindle between the two plain bearings fitted with a cone of three gears. Drive from these gears was picked up by a small gear mounted on a hinged shaft arranged in front of the headstock. By sliding the shaft left or right, any one of the three gears could be selected, with the drive transmitted from the small gear to a larger one. On the left-hand end of the front-mounted shaft was another gear, this engaging with a second set of changewheels that drove a leadscrew mounted centrally between the bed ways.
If the headstock and tailstock had been in their normal positions, this would have put the front-mounted gear-carrying shaft well forward and have created a serious safety hazard. Even though the lathe would have been built in the 1920s, or even earlier, when a lack of guards over belt runs and gears was not uncommon - in this case it must have crossed the designer's mind that having exposed gears so near to the operator was very unwise.