Edward —— The Luger is a 1916 DWM, and I believe the reason it needed a new barrel for police service was that it was an artillery model. It has the notch at the receiver/ barrel interface for clearance of the artillery sights, and it looks like the toggle piece with ordinary sights was added during the rework because the “matching” number is done in a different font than the rest of the numbers.
What I first thought was an eagle 35 barrel mark is indeed an eagle 33 as you suggested. Does that mark identify the maker of the barrel and the time frame of it manufacture?
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