Sounds like the good old Tommy Cooker. I don't think it would be wise to fold it any flatter than it already does.
There issued to us with our ration packs down here in Australia.
They do tend to rust out after a lot of use. Bit tricky to balance a yank style cup canteen on them but we have seen a hexi stove that fits to the bottom of the yank cup canteen that does a good job.