Hello,
just a bit surprised that no one pointed you over to
Hytta Antik, which is a trading site for antique lamps and replacement parts, run by the same operator as this forum. You can browse the parts posted for trade or sale there, or if you don't find what you need, post your own "I'm looking for" advertisement.
As to the actual designation of your burner, I'm afraid I can't help you there as I haven't studied "indoor" burners in to that detail yet. Some hints to find out more would be:
-what "gauge" is your burner? - i.e. diameter of gallery thread, inner and outer burner tube, width of the wick? If you're lucky this might even be stamped somewhere on it, e.g 10''' ("lines", denoted like "Inch" but with a third apostrope is an old unit of measurement which was/is used for classifying these lamps).
-how is the air induction system of your burner realised? (probably an "inverted-V" shaped aperture on the other side of the wick tube)
-is there a provision for a flame spreader (which would then be missing in your picture)? This would most likely be a small tube (1-3mm diameter) in the very center of the burner (inside the inner wick tube) which takes a mushroom- or funnel-shaped pin that sits above the wick. Other designs do however exist, like a tube of perforated sheet metal with the "hat" on it that sits on a ridge inside the burner.
Hope this helps,
Arno