A special issue of "Automobilia", a french classic car magazine, which covered in the same volume the 1927 and 1977 Paris Motorshow, and made by the famous journalist, René Bellu.
He seems to be very precise about that car and the fact that it was first shown in february 1927... I've looked in the old Georgano Encyclopaedia, and the Elgé car was not yet featured.
Moreover, I got another source, with the Burgess Wise/Cole Encyclopaedia. It says that "the Elgé brand was created in Bordeaux by Roger Louis Maleyre, pioneer of aerodynamics. Very light, low and thoughtfully streamlined, the Elgé cars used CIM engines. Maleyre built too a prototype using a velocity joint. Total production was roughly 30 cars."
And there is another picture, showing clearly the same car featured in that puzzle, so I still think this is the car we're talking about...