This rifle from the reference collection is a Italian Vetterli-Vitali M 1870-87. It was made in 1880 as stamped on the left side of the barrel. It is a magazine conversion of the M1870 single shot rifle. This conversion consisted of a metal shroud around the magazine housing, removing the dust cover and installing a knurled knob on the bolt to close of the magazine to allow single shot firing. There were a few other changes. For furtherinformation on this and other Vetterli rifles check out this very interesting book by Robert Wilsey.
Now on to the pictures