Welcome to the forum

I wasn't aware there was an M version. Where did you purchase it?

In the US doesn't help that much. From whom did you purchase it? The i337 is an AT&T model and not an International version of the phone. It it an S4 mini?

And I disagree. Look at the chart on this page. Samsung Galaxy S4 - Wikipedia, the free encyclopedia you will find that the ONLY carrier listed for the i337 is AT&T. i545 is Verizon and so forth. S4s beginning with a 9 are as listed.

I really would like to know why you seemed to insist that my statement re: the i337 was incorrect.

I'm trying my best to figure out if you have been bamboozled.

None of which seem to indicate that it's a US based phone and my comment specified that the i337 is an AT&T model, which is still correct.

Even if nobody knows what it is, there must be some difference. So getting back to the OP I would say that I would not even attempt putting the i337 firmware on an i337m. That's asking for potential problems. I saw some stuff about rooting that was for Canadian models only.

I still, after all these posts, (I may have missed it) can't figure out who the carrier was supposed to be as the model numbers other than the International, (unlocked) numbers are pretty carrier specific. No answer as to who the phone was purchased from AFAIK.

There are some ROMs that are compatible with different models of some phones. In the end there is usually a function or feature you would give up if it wasn't specific to that phone and even then getting most ROMs is a give and take on a senor or other feature depending on the phone. The same thing applies with some tablets. If you go down that road, be certain you have a recovery plan you fully understand.