A Brief History of the Marque
Manufactured: E. Allain
7 rue de l'Ane-Vert & 9 rue de Baboeuf, Kremlin-Bicêtre (Seine)
Possibly associated with Alphonse Thomann
Moto Revue of Dec 1st 1925 has an advertisement for A.X.A which claimed that the motorcycle had a gear driven magneto, chain final drive and drum brakes front and rear, with unusual leading link front suspension.
Cyclememory lists two models: 250cc Standard with Staub 3 speed and a 2T engine, and a similarly configured Confort model with balloon tyres. An image posted by Zhumoriste shows both the AXA and an Alphonse Thomann, identical but for the logos. The A. Thomann is shown as having a Train engine on both models, with belt drive on the Touriste. The Sport has a Sturmey-Archer 3-speed gearbox, brakes front and rear, and chain-chain transmission.
It is believed, based on the frame numbers in documentation, that over 200 machines were built in 1925-26. A surviving unrestored example fits the description of the advertisment above other than the front forks, which are conventional girder style.
Sources: cyclememory.org, zhumoriste.over-blog.com