Detective Sergeant Swinton said the woman had told police of her intention to kill both children, and also allegedly told witnesses the night before that she "had to drown my babies".
A woman accused of trying to drown her nine-year-old son is expected to also be charged with murdering his younger brother after police recovered the body of a five-year-old boy from the Murray River on the NSW-Victoria border.
Five years old, he's barely lived, ' one woman said after she and her young daughter placed a bouquet of flowers on a chain metal fence overlooking the river, off Perricoota Road.
Police launched a major search operation involving the SES and air wing, hunting for him all night, but he was not found. Emergency services expect to restart the search at first light Friday.
A Saturday morning court hearing heard the mother is likely to be charged with more offences.
The 27-year-old will appear in Deniliquin Bail Court tomorrow after being extradited from Victoria to New South Wales.
Further charges are expected to be laid at that time.
Brooks said government agencies had "culpability" for what happened to the two boys on Thursday.
It is alleged she then held him under the water in an attempt to drown him.
The woman walked to the river in Moama, on the NSW-Victoria border, with her children and told her eldest son, aged nine, to get in the water before she held his head under, police told Bendigo Magistrates Court on Friday afternoon.
He was taken to Echuca Hospital and transported to Melbourne for treatment.
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Dale Brooks, director of law firm Brooks Hall Lawyers, told the Sydney Morning Herald he'd made "considerable representations" to Family and Children's Services and Corrective Services about the "imminent risk posed to the little boys with their mother's release from prison", and "both departments failed to act and consequently we now have a grieving family". He was last seen drifting face down in the river, the court heard.
"The mother in this instance has been a really troubled individual, she's been a long-term ice addict", Mr Brooks told News Limited.
"She's pretty upset", the Legal Aid duty solicitor said outside the court.
Detective Sergeant Swinton told the court the younger son was screaming during the incident. "He requires some surgery for the dog bites and of course he's distressed; he's a nine-year-old boy". "We want answers", she said.
A neighbour outside the court on Saturday, said it was impossible to imagine what the grandmother of the children was going through.
Deniliquin police superintendent Paul Condon told reporters the case was "just terrible" and that it "will rock the community of Australia".
But there's nothing but sympathy for the 27-year-old's mother and stepfather - the grandparents of the child found dead in the river and the other in hospital in Melbourne.
Police said the woman was arrested and she was helping with their inquiries.
She was released February 2 this year. "The actual circumstances are unknown", Supt Condon said.
The incident happened at Moama, near the Murray River.