The OF mark is actualy DF which stands for Distrito Federal (Federal District of Mexico City). The Eagle mark was used roughly from 1948 to 1980. They changed over the years. Your mark looks like an older mark to me. Don't know the manufacturer who used the BGM mark.
A few years ago I had an opportunity to buy two very old Mexican silver books with many maker's marks inside. The books were in rough shape so I passed up the deal. I'm just now realizing how rare those two books were.
Thank You! Every bit of informatrion helps. You are so right about the books. I try to get my hands on all the information I can. It is frustrating not knowing what one has....Thanks Again. It is appreciated!!!