Back in the 70’s the datejust was available on a president bracelet. This maybe even earlier. Looks gen. Do you not have an internet search engine? It takes 30 secs to look! just sayin. Never seen that clasp before though. From a grainy picture it’s impossible to tell.
Rolex never made a president style bracelet in 19mm that would fit a 34mm Date or Oyster Perpetual, which were made in 14K solid gold. There are a number of aftermarket 19mm president style bracelets available, mostly from Italy. Rolex made an Oyster with rivets and a Jubilee both with oval and then 'D' shaped links.
The fact that the maker of your pictured bracelet is trying to pawn it off as a Rolex product based on clasp engraving should be a tip off as to the integrity of the seller or owner, if this isn't for sale. You can also see the sloppy fitment at the end link/case point of contact.
That clasp looks real, South American from the engravings. But the bracelet links are questionable. Either fake or from another bracelet. You would need to check the gold content. The fit to the case is horrible. The end links are for a much larger case and may have been ground down to fit between these lugs.
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