What would happen if a person had amnesia and could not be identified?

This is for a story.



If an adult was 'found' with no memory of themselves or who they are and the police were unable to identify them, and no one stepped forward to claim them, would they be 'let go' or would they be commited? This is assuming the they had all thier other faculties and intellegence except for thier indenty.What would happen if a person had amnesia and could not be identified?
Actually, they would be able to identify them through DNA testing or fingerprints. Since it's for a story though, just make something up. The authorities wouldn't have any grounds to detain the individual, so they'd be let go on their own.