The paling to the right surrounded by my garden have blown down. Who is liable for its repair. Is it both neighbours? ...

Is near a passageway you can save a...
Answers:    are you in the UK?

If so consequently you should have be told when you purchased the property... if you dont have this info consequently the next best approach you can find out which boundaries you maintain is by looking at your deeds documentation (if you dont own them then you can carry a copy from the land registry bureau for a small fee).

If these don't show it, and they might not, then it would be down to who have been maintain the boundaries - they would continue to be liable.
Drawf wicker?
Who ever owns that fence/property is responsible for it.