I would read the initials as "J J" despite not seeing the top of the second letter clearly. The guard's small rear spur makes this gun a little harder to place for me. The top view of the butt plate extension doesn't show a lot of width, and the single lock bolt with what appears to be a well-fitting percussion lock, along with a very simple cheek shape, all seem to suggest a circa 1835 gun to me... not sure how we are seeing this gun as much earlier... unless I'm missing something. The square end on the percussion side lug and straight comb and toe lines often suggest a southern rifle, as has been suggested.
Shelby Gallien