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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. A pair of images 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