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There is no statutory instrument and Article 50 could be invoked by any single person in the UK.
However, The EU is unlikely to accept notification from anyone other than the Prime Minister or duly appointed successor or sufficient remaining authority. They are also unlikely to accept it if they feel that the notification lacks sufficient backing of the voting population or parliamentary representation.
Under current circumstances, the second condition has been met.
A proper vote, in Parliament.
Does any one person have the authority to unilaterally invoke it themselves? Is there a statutory instrument, or does there have to be an Act passed?