Ranjeeta Sharma's husband, courier driver Daniel Sharma, flew to Fiji with their 4-year-old son on Friday, hours after firefighters found her burning body on a road west of Huntly in the Waikato.
The case has parallels with that of Nai Yin Xue, who killed his wife An An Liu and fled New Zealand to Melbourne in 2007, where he left his 3-year-old daughter Qian "Pumpkin" Xue at a railway station.
Mrs Sharma - a 28-year-old nurse who worked at Middlemore Hospital - was involved in a violent marriage and moved out of her Manurewa home recently, after complaining of being beaten by her husband.
Her Facebook page contains photographs of her son.
"My little boy, my life," she wrote under one photo.
Neighbours of the couple confirmed she had moved out before Christmas.
They also said they owed considerable amounts of money and had been visited by police and debt collectors at different South Auckland addresses.
Mr Sharma left his wife's silver Subaru station wagon at Auckland Airport on Friday. The car is being forensically tested and police have appealed for sightings of Mrs Sharma, or the car, in South Auckland or the Huntly-Rotowaru areas last Thursday or Friday.
Mrs Sharma was alive when she was set on fire.
She could only be identified after a post-mortem examination.
"A key focus of the inquiry now is establishing Ranjeeta's movements and how she came to be where her body was found," said inquiry head Detective Senior Sergeant Nigel Keall.
He said the husband was "a person of considerable significance" to police, who were working with Interpol and other authorities to find him.
A report last night said Mr Sharma was believed to have a Canadian passport, among others.
"I would like to reiterate that while we are seeking to speak to Ranjeeta's husband, we are keeping an open mind as to what, if any, involvement he has in this matter."
He said search warrants were executed at several addresses yesterday.
A number of the investigation team had moved from the Waikato to South Auckland because that was where Mrs Sharma lived and worked as a nurse at Middlemore Hospital.
Neighbour Des Leota (30) helped her move to Papatotoetoe last month.
He said yesterday she wanted to leave the family's Manurewa home because he said her husband had beaten her.
"She called the cops and they took him away to the cells for the weekend."
Despite her claims of abuse he'd never heard them arguing but did see her one day with a black eye.
While he was gone, she took her chance and left the house, with her son.
"She seemed scared but [after the move] she seemed happier."
Mrs Sharma told him her husband moved to his sister's home in Winsford St, Manurewa, after they split.
That house was yesterday searched by police and forensic investigators.
They paid special attention to a car parked in the driveway and carefully examined the boot.
When Mrs Sharma moved, she sold many of their possessions because she wanted to make a clean break, Mr Leota said.
"She didn't want him to follow her."
But he was amazed days later when she returned to the house to clear out more belongings - with Mr Sharma. "After all that, she shows up ...
"She must have gone back to him."
The couple was in her grey Subaru station wagon that was found abandoned at Auckland Airport.
"I thought they must have worked things out."
