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".
Police were called to Perricoota Road near the Murray River after a nine-year-old boy was found with what police said were "serious dog bite injuries".
She was then last seen with her five-year-old son in the river, he said.
The search operation was launched involving police from Deniliquin Local Area Command, SES, and Maritime Rescue, and the Victorian police airwing.
Crews had been desperately scouring the River for a second day in the search for the boy, while his mother faces an attempted murder charge.
She appeared in a Bendigo Court in Victoria on Friday afternoon with NSW police extraditing her across the border to Deniliquin, where she was arrested and charged with the attempted murder of the nine-year-old.
"We anticipate there will another charge", the police prosecutor told the court.
The nine-year-old underwent surgery in hospital, and is now in a stable condition.
The woman turned herself in to police in Echuca, on the NSW-Victoria border, on Thursday night after she was apparently attacked by a dog at the scene of the incident.
The grandmother's lawyer, Dale Brooks, said he had made "considerable representations" to both Family and Children's Services and Corrective Services about the "imminent risk" to the boys. He was last seen drifting face down in the river, the court heard.
The accused is said to have been living in a caravan at the back of the grandmother's property after her prison release, despite a 12-month Apprehended Violence Order (AVO) taken out against her by the grandmother in May 2016.
"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 had reportedly told witnesses she "had to drown my babies". However, she kept being told there was no where for the woman to go.
Detective Sergeant Swinton told the court the younger son was screaming during the incident.
Despite the grandmother taking out an apprehended violence order against the woman previous year, authorities still expected them to live together. "We want answers", said the woman's aunt, who can't be named for legal reasons.
Shortly after, Brooks said the woman took the grandmother's auto, snatched the two young boys and left with a couple of her buddies.
"Without any communication by your office (or police) or agreement from our client, (she) was released from jail. and taken to our client's home with no regard for the AVO in place protecting our client", it read.
"I think they (the agencies) have some culpability in this case".
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. "The actual circumstances are unknown", Supt Condon said.
Tensions heightened after Prime Minister Narendra Modi in February threatened to block the flow of water into Pakistan. They said that India bought time to complete the two projects and then said that the projects could not be modified.
The Samsung Galaxy S8 will go on sale April 21, following an official unveil in NY on March 29, the latest leaks indicate. Apple is believed to have removed the 3.5mm audio jack on the iPhone 7 and iPhone 7 Plus for two reasons.
He said the insightful comments and thought provoking views of the participants had indeed added value to the deliberations. Uzbekistan's deputy prime minister will attend the event. "We need to work on streamlining our institutional mechanisms".
Meanwhile, Snap's chairman Michael Lynton planned to sell nearly 55,000 shares in the offering, bringing in more than $930,000. Snap said earlier that it will use a small portion of the proceeds for settling tax obligations related to stock options.
The president stated before an audience of sailors that his budget proposal to invest $54 billion into defense spending. Ford carrier, which has been under construction since 2009, is expected to be commissioned this summer.
Wagner remained upbeat about his side's trip to the Etihad following the two sides' goalless draw at the John Smith's Stadium. The Belgium worldwide has played only six times this season, and has been dogged by injury woes over the last two years.
The sale was disclosed in a legal filing with the Securities & Exchange Commission, which is accessible through the SEC website . The share price is now 0.48 percent versus its SMA20, -1.8 percent versus its SMA50, and 22.48 percent versus its SMA200.
In addition, he said identifying carbapenem-resistant bacteria as the highest priority for research and development was "spot on". Each year, at least 2 million people acquire antibiotic-resistant infection and 23,000 die as a result, according to the CDC .
When asked by a fan if we might ever see a black-and-white cut of the arty X-Men film , Mangold said he's working on it . But what about the main X-Men films? X-Men: Apocalypse launched in IMAX when it premiered past year .
It wouldn't be shocking for both he and Fournette to be available at No. 8, but it also wouldn't be shocking if Cook went first. That attitude doesn't surprise McCaffrey , but it sounds like he doesn't think that should happen under any circumstances.
He said that Russian Federation , which along with Turkey helped broker a ceasefire in December, appears to be softening its stance toward the armed opposition.
Polling begins in 38 constituencies for first phase
Playing the saffron party's election song, "Thambal", the bus travelled in various parts of Imphal East and Imphal West districts. As per the 2012 elections results in these 49 seats, 27 are won by SP, 9 by BSP, 7 by BJP, 4 by Congress and others 2.
Trump calls for probe into Schumer over old Putin photo
But given White House opposition and Republican control of Congress, he said none of the options seem likely at the moment. Schumer is among the lawmakers pushing for an investigation into the relationship between Trump and Russian officials.
The Legend of Zelda: Breath of the Wild Began in 2D
Zelda: "Breath of the Wild is an evolution of the formula for both eastern and western philosophies alike, and a new blueprint". Well, so did Nintendo's Satoru Takizawa, who oversaw art direction for the latest entry in the series, The Breath of the Wild .
Snapchat owner's stock leaps 41% on U.S. stock market debut
While Snap's IPO boosted the net worth of other Snapchat executives, it pales in comparison to the founders' newfound riches. Facebook's first day on the market showed higher stock prices ($42.05 per share) but came back with an 11 percent increase.
Senate confirms Perry to head Energy Department
Sonny Perdue, the former Republican governor of Georgia, is Trump's pick to oversee the Agriculture Department. Hoeven said he looks forward to working with Perry to provide a true path forward for clean coal technology.
'Dancing With the Stars' season 24 cast revealed
If you remember way back to when Heather first rose to fame - she competed on Season 2 of So You Think You Can Dance . Former New York Giants running back Rashad Jennings will fill the annual National Football League player quota.
Russia Bombed US-Backed Coalition Fighters in Syria
Townsend told reporters that he believed the Russians and Syrians had thought that the villages were held by Islamic State forces. Earlier Wednesday, a USA commander accused Russian and Syrian jets of bombing US-backed fighters in Syria.
Mount Etna eruption captured by drone camera
Mount Etna, Europe's most active volcano, spews lava during an eruption, near the Sicilian town of Catania, southern Italy. Etna can still cause accidents with its ash cloud, so officials say that they are tracking it.
Iran tests missiles in naval exercises
Gulf allies have long favored a tougher U.S. stance against Iran, whom they blame for regional interference. Another laser-guided missile, Dehlaviyeh, had also been successfully tested, reported Tasnim news agency .