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Regina were motorcycles produced by the Magneto Motor Manufacturers of Lewisham, London between 1906 and 1907
1906-1907 The firm produced a basic model with a loop frame. They also imported the Belgian CIE models for a short time.
See Ilford Motor Car and Cycle Co for more likely connection
The British Motorcycle Directory - Over 1,100 Marques from 1888 - by Roy Bacon and Ken Hallworth. Pub: The Crowood Press 2004 ISBN 1 86126 674 X