Police are investigating any online mentions of the woman charged with murdering two children in Northland last week, who by court order cannot be named.
RNZ has seen breaches of the order, which name the woman.
Northland CIB detective inspector Bridget Doell said police were making further inquiries into some activity on social media.
"Police are continuing to monitor the situation with regard to suppression breaches around the Ruakākā homicide," Doell said.
"I acknowledge the public's feeling about this tragic case, however I must remind the public that it is an offence to breach a court order, and this includes naming someone who has name suppression on social media."
Doell said if anyone were to come across a post in breach of the name suppression order, police strongly advised them to remove, not share, such posts on their account.
The woman was remanded in custody without entering a plea in Whangārei District Court on 16 May.
Whangārei Judge Gene Tomlinson granted the woman and victims temporary name suppression, including their ages.
Police launched a double homicide investigation a week ago, after finding two children dead at a house on Peter Snell Road.
Doell said police would not comment further now the case was before the courts.
Anyone who breaks the law in New Zealand could face a fine of up to $25,000 or six months in jail.
In 2021, Auckland businessman Leo Molloy was sentenced to community work for naming backpacker Grace Millane's killer while a suppression order was in place.
Molloy, who had pleaded guilty, was also ordered to pay a fine of $15,000.
Millane was murdered by Jesse Kempson, who she met on Tinder in December 2018.
Kempson's name was suppressed by the courts until December 2020, but was published in international media outlets.