No one has been charged with JJ's horrific death to date, but a police source said the homicide team were determined to "keep shaking the tree" until they got the information they needed to make an arrest.
Detective Inspector Scott Beard said officers were still interviewing the residents and associates of the Takitimu St house. So far, the adults present when JJ died had given explanations "inconsistent" with the results of the post mortem.
Miss Lawrence moved into the house with JJ in July. Before that she had been living with the little boy, who would have turned 3 in March, in Tauranga.
Police are now urging her previous neighbours, friends or associates to come forward if they have any information that might help them find out what happened to JJ.
The Herald revealed yesterday that Miss Lawrence was pregnant with her second child and Child Youth and Family may be considering removing the new baby from her custody after its birth.
Police said Miss Lawrence went to Northland for JJ's tangi last Thursday and had not returned to Auckland. It was unclear when she would return, but police were confident they would speak with her soon.
"Police have not been able to speak with JJ's mother since November 14 but hope to do so in the coming days," spokeswoman Noreen Hegarty said.
They confirmed Miss Lawrence's partner is currently in Auckland, but not living at the Takitimu St house.
A neighbour said he went to the house several days ago, "packed all his stuff and took off".
Ms Hegarty said police had contact details for all of the adults present at the house when JJ died, and while some of them were no longer living there - no one had "fled".
She said police could not comment on the specifics of the investigation for "operational reasons". But they wanted to assure the public that their investigation was already "robust" and would be "thorough".
JJ was born at Auckland Hospital in March 2009. His father James Robert Ruhe was said to be "devastated" and "lost" when told of JJ's death.
Ruhe is currently in prison and did not seek compassionate leave to attend the toddler's tangi near Kaeo. Instead, the Department of Corrections has allowed him to transfer to a prison closer to his family in NOrthland, so they can support him as he grieves.
Ruhe will remain at the temporary prison until the New Year and a memorial service will be held there for him and his family.
*If you can help police, contact the Operation Avalon team on 09 524 1925 or call Crimestoppers anonymously on 0800 555 111.