Attached are new pictures of the pistol, they show much better the very nice original finish.
Reading up a few internet sites and comparing with other pistols I did notice the slide inscription on my pistol reads "FÈMÀRÙ", so also ´ on the last letter U. This is only found on the very early 37M military pistols, as well as the very last 29M pistols, the ancestor of the 37M. Later they changed the inscription to "FÈMÀRU". Based on this it is clear the pistol was made at the very beginning of the 37M production. But was this a commercial production or even the trials series...?
PS: found out among the 29M pistols there are also circled E accepted pistols, as well that this marking is said to also can be encountered on 35M and 43M rifles. So maybe a different military acceptance?