"This includes various speculative theories about the possible circumstances surrounding Mr Tucker's death.
"This speculation, while understandable, is extremely difficult for Mr Tucker's family, and is also not helpful to the police investigation," Mr Fitzgerald said in a statement.
"We would therefore ask that the media respect the privacy of Mr Tucker's family, and also carefully consider the veracity of any information they choose to publish or broadcast.
"While Police are keeping an open mind as we proceed with our investigation, our focus is on dealing with fact, not unsubstantiated rumour."
He said police would release information when it became available, where appropriate and asked for patience from the media and members of the public.
"This is a complex investigation and it may take some time before we are in a position to speak in any detail.
"We can say however that we do not believe there is any risk to the wider community."
Fairfax has reported that Mr Tucker was jailed in 2000 for 10 months, for charges including breaching hire purchase rules, cheque fraud and drink-driving while suspended.
In 2002, the former landscaper was sentenced to community work following his third drink driving offense when he crashed off a road in Oxford and hit a tree stump.
Then in 2006, police were looking for him after he breached his release conditions.
Mr Tucker's body was found near lake Lake Ellesmere on Drain Rd, Doyleston on Monday, by a group of Filipino fishermen.
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• Police identify body found in rural Canterbury