The woman was named today as Ranjeeta Sharma, 28, of south Auckland, whose body was found burning by the side of Hangapipi Rd in the rural township of Rotowaro, near Huntly, on Thursday.
Police said she was alive when she was set alight, and that an accelerant was used to start the fire.
Speaking at a press conference in Manukau today, investigation head Detective Senior Sergeant Nigel Keall said the woman's husband was "a person of considerable significance" in the investigation and police were working with Interpol and overseas authorities to find him and the son.
Her husband, a Fijian national, flew to Fiji with their son the day after Ms Sharma's body was found. Her car was found abandoned at Auckland Airport.
Police were not naming, or releasing any details about, Ms Sharma's husband or their relationship history.
Mr Keall said they wanted to speak to the husband but were keeping an open mind about his involvement in the death.
New Zealand has an extradition treaty with Fiji. A spokeswoman for Fiji police told NZPA they were aware of the case but would not comment further.
Police here were forensically examining Ms Sharma's silver Subaru station wagon, registration FSD433, and appealed for people who may have seen it or Ms Sharma in the Manukau or Huntly-Rotowaro areas on January 20 or 21 to contact them.
Mr Keall said a large proportion of the investigation team was brought to south Auckland because that was where she lived and worked.
Police were executing search warrants around south Auckland today.
Ms Sharma had at least one close relative here but most of her family was in Fiji.
"Ranjeeta was burned to death beside a public road so obviously her family are in shock, it's been particularly difficult for them," Mr Keall said.
Television reporters went to Ms Sharma's previous flat in south Auckland, which the current tenants said was left in a mess. As well, people had visited the flat demanding money, and letters had arrived for Ranjeeta Sharma and also Ranjeeta Rekha, whose Facebook page lists her as working at Middlemore Hospital.
Neighbours said the couple were involved in a courier business but left suddenly, possibly because they owed rent.
Amnesty International said that if Ms Sharma's death was an honour killing, it would be the first in New Zealand, TV3 reported.
"Honour killings nearly always occur because women are perceived to be property and across culture, across religions, they are seen as property and you can do what the hell you like with them," Amnesty's Margaret Taylor said.
Amnesty had no record of any honour killings here or in Fiji but said domestic violence was high among all cultures.
Auckland Indian Society president Harshad Patel said media speculation that it was an "honour killing" was premature and it seemed unlikely to be the case.
"In civilised societies you don't see this sort of thing, especially in New Zealand."
