Police have confirmed that Marcus Luke Tucker was brutally assaulted prior to his death, and his body was burned.
The 36-year-old Christchurch man's body was found dumped near a popular lake in rural Canterbury on Monday.
Detective Inspector Tom Fitzgerald said Canterbury police were continuing to provide support to Mr Tucker's family as the investigation progressed.
"We are committed to finding the person or persons responsible," he said.
"Police are aware that there is a lot of interest in the local community and media about the circumstances of Mr Tucker's death, and that this is prompting a certain amount of rumour and speculation.
"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. "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."
Mr Tucker's body was found near lake Lake Ellesmere on Drain Rd, Doyleston on Monday, by a group of Filipino fishermen.
