It's nothing to do with the council, unless the house next door is owned by them.
Land registry keep copies of deeds.
The deeds will only show who is responsible for the boundary, (should be marked in red) either side could have put up a fence within the true boundary.
If the fence is ON the boundary the best side, if there is one, normally shows to the person who is NOT responsible.
If the house hasn't been sold in the last 30 odd years then the land registry details may be incomplete.
Boundaries are a nightmare.
I bought my house new, and know exactly who is responsible for what.
My boundary to the left is next doors, boundary to the rear is another neightbour, half the boundary to the right is one next door neighbours, the other half is joint between me and a different neighbour as it is a brick wall built when new.

You will have to speak to your neighbour to see if they know whether it's on their deeds and come to an agreement if not.


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