A 27-year-old Riverina woman has been placed in Junee Correctional Centre and will face court on Monday, March 6.
The nine-year-old remains in a stable condition at hospital in Melbourne. He reportedly suffered serious dog bites during the incident on Thursday afternoon.
The mother handed herself in to police at Echuca on the Victorian side of the river about 10pm on Thursday, police said.
The woman had allegedly told onlookers "I had to drown my babies".
The mother-of-two chose not to attend the local court house in Deniliquin, where she was extradited to from Victoria on Friday evening.
A 27-year-old woman is expected later this morning to face a charge of attempted murder.
During the court hearing, she was refused bail. She will appear in the Deniliquin Local Court tomorrow.
"Further our client did not want her daughter to be living at the property since she was a victim of her daughter's conduct over many years".
The boys' grandparents are furious at government agencies, believing they failed the family and could have done more in the lead up to the tragedy, said Dale Brooks, a lawyer for the family.
He said government agencies had "culpability" for what happened to the two boys.
"The mother in this instance has been a really troubled individual, she's been a long term ice addict".
The search for the boy resumed on Saturday morning with oolice and emergency service boats back on the water and divers expected to pick up where they left off at dark on Friday.
Keith Thurman Vs. Danny Garcia Thurman fight will finally get settled in Brooklyn. " This fight reminds me of my incredible  bout against Tommy Hearns". It was a natural fit and Thurman won the WBA welterweight title that year with a 10th round TKO victory over Diego Chaves.
"A number of organisations ought to be held to account", Mr Brooks told the ABC.
"It's an appallingly sad country town saga", Mr Brooks said.
Superintendent Condon said there had been "a couple of incidents" with the family since the mother was released from jail.
Brooks said the woman took the grandmother's vehicle on Wednesday night, snatched the two young boys and took off with a couple of her buddies.
It's understood the mother, who doesn't have custody of her sons, was released from prison recently after being arrested late past year on two outstanding warrants.
The court also heard she had allegedly told witnesses she "had to drown my babies".
Detective Sergeant Trent Swinton told the court the younger son, aged, 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".
'The dog has been involved.
Police allege their mother, who can not be identified for legal reasons, held both their heads underwater, but the older boy managed to escape.
"There is absolutely no link between Justice Health and the public health system in the community", she said.
The tragic discovery comes after the five-year-old boy went missing in the river at Moama, in the NSW Riverina region, on Thursday.
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The people who came here are here because they made bad choices and they're violent. "That's why they're in prison". All of the inmates are in the general population of the prison, but are being held in the highest security.
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