A Brief History of the Marque
Alphonse created a firm in his own name in 1924, but the use of it was curtailed after a court decision.
Bicycles were built under the Alphonse Thomann marque, and motorcycles were marketed but were possibly rebadged models manufactured elsewhere. A document dated 1923 mentions "fourches elastiques pur velos et motos", and gives his address as Avenue de Fouilleuse, Suresnes (Seine). This was apparently before the company was created.
At the 18th Cycle Show Alphonse took a stand right beside that of Thomann. Edmond Gentil, who had bought the Thomman name from Alphonse and his brother Jean, was far from delighted and took Alphonse to court. By 1926 Alphonse had left the two-wheel trade and began building refrigeration equipment.
Zhumorist refers to the episode thus: "tangled stories of lawsuits, formal notices, prohibitions, property rights, etc." His article on the subject states that a 1924 175cc two-stroke was produced, and that the following year an identical machine was presented under the AXA marque, showing clear images of both from one of his books.
Sources: wikipedia.fr, encycloduvelo.fr, babethhistoires.centerblog.net, Zhumorist.