You mean: What is a "1937 Alleggerito?"
A 1937 commercial Beretta with a lightened barrel is somewhat unusual. It's apparently a 1935 (.32 ACP) with lightening cuts on the barrel.
Try this link for info: https://www.proxibid.com/asp/LotDeta...88&lid=8488272
I never heard of one without a serial number on the frame. Make sure you don't have one with an altered/defaced S/N.
If, in fact, it left the factory without a serial number it might be a prototype, since, from what I came up with, it was more or less an experiment by Beretta.
So, if by chance you have encountered one that never
had a serial number applied, you are still legal, since the gun was made prior to 1968.
Make 100% sure that the gun NEVER had a serial number, though.