New Zealand Law
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The neighbour also has obligations to ensure his structure does not collapse on to your side. So the engineers report will enable you to have an informed discussion about this with the neighbour.
If the neighbour and the council do not do anything, then you can instruct your lawyer to place them on notice that you will sue them for any damage which results.
I realise you have spoken to them already, but presenting them with hard evidence is much more compelling