Hi Martin,
Sturmey Archer made the LS gearbox from 1922 to 1930 (3 speed with the number prefixed either "LW" or :HW").
There are a large number of variations to be found because they were made with different casings, diffferent kickstarters, different mainshaft lengths, etc, etc, to suit customers requirements.
You need to say what bike you want the box for.
The Vintage Motor Cycle Cl ub Library holds a lot of original Sturmey Archer literature and you need to go through it closely to work out what suits your bike.
I got to know Sturmey Archer boxes quite well through my interest in vintage Matchless. Matchless used Sturmey Archer boxes on all their models except one thoughout the 1920's until Sturmey Archer stopped making bixes around 1933 or so.
They are easy to work on but the number of variations found makes life a bit frustrating at times.
Cheers, Bob