Detectives refused to say what had happened to Miss Grenside, 25, but confirmed they were not looking for anyone in connection with the incident.
Detective Superintendent Ian Whinnett, of Hertfordshire police, said he was satisfied he was not looking for anyone in relation to any offences committed against Joanna.
He said it was 'very likely' a file would be submitted to the Director of Public Prosecutions when his inquiries were complete.
Miss Grenside had no motive for inventing the abduction story and was not with someone or somewhere she should not have been, police said. 'We can only assume it was her mental state at the time. We have found no motive behind her story.'
Suggestions last night included the theory she suffers from the eating disorder bulimia nervosa and could not face festive eating and drinking. Police last night asked commuters at Harpenden railway station if they had seen her taking a train to London on the night she vanished.
Det Supt Whinnett said: 'As far as we can see there was nothing malicious behind this and it is all very unfortunate. Joanna is still very distraught and is being looked after by her parents.'
The family were not answering calls at their home in Harpenden, Hertfordshire, yesterday.
Miss Grenside was reported missing last Tuesday after she failed to turn up for an aerobics class at 7pm at Harpers Leisure Centre, in Harpenden.
Later her rape alarm was found near her locked car, left in the centre's car park.
She reappeared, distressed, two days later, saying she had been abducted by two men. She claimed she had been blindfolded and held for two days in an isolated house. Her disappearance led to a massive search.
Specially trained officers were unable to question her in detail about the story because she was said to be too distraught to talk, but officers interviewed her at length yesterday when she admitted the story was a fabrication.
Det Supt Whinnett said that she gave police few details about her abduction and said she had not taken off her blindfold, even though she was left alone for up to 12 hours, as she was too frightened.
Police maintained throughout they were 'keeping an open mind' about the case.
At the weekend they said they were investigating similarities with a faked abduction in Australia.
In that case Fairlie Arrow, 27, a singer, went missing for two days and was found tied up in a road in Queensland.
She claimed she had been abducted by a fan, blindfolded and bound to a four-poster bed, but two weeks later admitted making up the story to boost her flagging career.
The incident happened exactly a year ago and police discovered Miss Grenside was in Australia at the time.
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