That looks like a Westley Richards 1871 patent. It is sometimes referred to as the Westley Richards Improved Martini. It is not really an improved Martini, the original model being patented 40 days before Martini.

Hector this photo with the number 38 says the caliber of the bore is 38 lead ball to the pound or about .525 to .526. This seems to point to the the 577/500 2 3/4 No.2 Express.

You need to get a copy of Wal Winfers book British Single shot rifles Volume 4 Westley Richards. Your rifle is serial number 503. It is not on Wal Winfers known single barrel list. You have a rifle that previously wasn't known. Serial number 502 is listed as possible 1873 Pattern. There are several No. 2 EX listed for the Improved Martini and there is one 1871 IM listed as a 577/500 No. 2 Henry rifling