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As to ownership of the property you can submit an index map search at the Land Registry attaching a plan to the form:-
They will reply letting you know the title number (if any) to the land. You can then download (or request by telephone) the official copies from the Land Registry. This will show who actually owns the land.
You should also check your records for when you purchase the property as there may be covenant for you to contribute to the repair of the strip of, possibly in common with other owners of other properties using the land. This information should have been provided to you by your conveyancer when you purchased the property. It will also be detailed in the official copies to your property (though you may have to also request a copy of a older conveyance listed on the official copies).
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