I zoomed in on your plane looking at the adjusting wheel. im guessing its an old millers falls. it has more of a diamond pattern like this one.
as far as the wavy stuff I think someone did that to the sole and the sides. I don't know of any advantage to that except to identify it, maybe from a great distance haha. just a guess. all my planes are old Stanley/bailey. heres a site to look at that might help
welcome to the forum. I envy you living in new Zealand. we had an exchange student from gore south island live with us. the pictures I have seen of your country make it a place I want to visit someday. then of course being from Oklahoma I want to thank you for steven adams
Humm ... interesting. I zoomed in as tight as I could and there are definitely letters raised in the sole casting, just below the thumb wheel. After rotating the photo a couple of times, and sitting with my head on one side whilst squinting I think it says "made in the USA" ... dunno, however there area couple of other clues in the link, so thanks for that.
I agree with you when you say the the waves are decorative, but I did think that it might have been a company trademark.
Yep, I moved to NZ from the UK about 15 years ago, and yep there is some stunning scenery around the place, however there are a few things about the place that I wish I'd thought more about before moving over here. The problems are, in my opinion, related to NZ's "isolation", because until recently it took a long time to get anything here, and when it did arrive it was horrendously expensive. The other detractor is the tiny population, and amongst other things that also increases prices because the demand isn't great enough to generate competition in suppliers.
On the other hand, I've got more true friends in NZ than I ever had in the UK. I mean the type of friend where you can knock on their door with no money in your pocket, and they'll feed and clothe you without a second thought ... :-)
As for Steven Adams, if you're looking for an Olympic Gold Medal winning female Shot Put, check out his little baby sister, Valerie ...
 In the link, it makes reference to Miller Planes manufacturing planes for other companies to brand / sell, and I think that could explain a few things.
Anyway, if I can get the plane for sensible money, I'll grab it and see what we have, thanks [\edit]