Fallschirmjaeger, my 1903 has the identical scope mount system as yours. Fortunately, the original rings and scope were still with my rifle when I stumbled across it. The rifle was proofed in 1939 and the mounts and scope (4X Hensoldt) are from that same era. The mounts are marked 'AKAH', but I think that was a retailer, not the mount manufacturer. I did have to get the scope cleaned up a bit, but Iron Sight, Inc., did a great job at a very reasonable price.
I don't know if your plans include having new rings made to fit it, but I can testify that the mounting system works like a charm! When first working up handloads and wringing out the rifle, I deliberately removed and replaced the scope many times just to establish a comfort zone about it's return to zero. It works very well. I imagine having rings made would be costly, but if your eyes are like mine, hunting without a scope is a very iffy thing. If mine had one of the bolt mounted apertures used to sharpen up the barrel sights, MAYBE I could pull it off. Unfortunately, that is not the case.
Hope you enjoy your M-S as much as I have mine. They are truly masterpieces of the hunting rifle world. Even handicapped with a scope as mine must be for it to be useable, they handle like no other!