Responsibilities for Chimney Repairs

Our neighbor’s chimney collapsed a while ago as a result of neglect.

The chimney is right on the party wall between the two brownstone buildings, but the flues are all my neighbor’s. (Our own chimney is facing the neighbor on the other side.)

My neighbor is under the impression that we are responsible for repairing our side of the chimney, regardless of the fact that the chimney’s usage is actually hers.

The chimney structure sits on the party wall, but the chimney breast protrudes more into her property than it does into ours.

Besides the cost for the repair, I am wondering about a potential hazard if any further collapsing would occur, because one of the flues is actively used for her (!) oil burner in the cellar.

Can anybody shed some light on the legal situation? Who is responsible for the repair?