Id say that the 37mm is indeed that, an AP or solid shot round from either the Germans, British or Americans in ww2, but I am guessing German? All the Allied ones I have seen on the web have 2 or more grooves at the bottom - driving bands??
I had a Foreman once who used a shell very similar to your 37, as a plumb bob!! It was a NZ one dug out of a berm at Burnham by his Uncle I think.
As for the case, if it is 137mm long then I guess a modern 25mm round such as fired by the bushmaster cannon in the Lavs
Cool though, I always remember being in awe of an inert 105mm round on Dads mates hearth as a kid.
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