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Pistorius, 26, one of the world's most recognisable athletes, was charged on Friday with murdering swimsuit model and law graduate Reeva Steenkamp in the early hours of the previous day.
He broke down during a 40-minute bail hearing at a Pretoria court but was not asked to enter a plea.
"They had plans together and Oscar was happier in his private life than he had been for a long time," Anthony Pistorius said in a statement released by his nephew's agent.
"We are in a state of total shock - firstly about the tragic death of Reeva who we had all got to know well and care for deeply over the last few months," he said. "All of us saw at first hand how close she had become to Oscar during that time and how happy they were."
The suggestion that Pistorius' family was close to Steenkamp runs counter to comments from Pistorius' father, Henke, who told the New York Times he had never met his son's partner.
"I don't discuss my son's relationships. I have, in fact, not met the lady," he was quoted as saying.
Prosecutors alleged the shooting was premeditated, a charge that could put Pistorius behind bars for life if convicted.
Police have said there are no other suspects and the pair were the only people in the house at the time. Initial reports suggested he may have mistaken her for an intruder.
Anthony Pistorius reiterated the family's belief that the track star - a double amputee who became one of the biggest names in world athletics when he ran at last year's Olympics - had not shot Steenkamp deliberately.
"After consulting with legal representatives we deeply regret the allegation of premeditated murder," Anthony said.
"We have no doubt there is no substance to the allegation and that the state's own case, including its own forensic evidence, strongly refutes any possibility of a premeditated murder or indeed any murder at all."
Pistorius is being held in a Pretoria police station before the resumption of his bail hearing on Tuesday. He is "numb" with shock and grief, the statement said.
Several South African media reports have said Steenkamp was shot through the bathroom door and was hit four times - in the head, hip, arm and hand.
Police said witnesses had heard disturbances at the home before the shots, adding that there had been previous incidents of a "domestic nature" at the home.
The shooting has shocked South Africa, where Pistorius was revered as a hero whose achievements transcended the racial divides that linger in Nelson Mandela's "Rainbow Nation" 19 years after the end of apartheid.
The disbelief was felt across the globe among the millions who saw in Pistorius the ultimate tale of triumph over adversity - a man who rose to the highest pinnacles of athletics despite having no lower legs.
He was born without either fibula but reached the semi-final of the 400-metres in the London 2012 Olympics, running on high-technology carbon fibre prosthetic 'blades'. He also won two Paralympic gold medals and one silver medal.
Although the public image was of a charming and easy-going young man, a Twitter posting by Pistorius in November paints the picture of a would-be action man obsessed with security.
"Nothing like getting home to hear the washing machine on and thinking its an intruder to go into full combat recon mode into the pantry! waa," read the Tweet on the morning of Nov. 27
Police recovered a 9mm pistol from his home after the shooting. The Afrikaans-language Beeld newspaper said Pistorius had a licence for one firearm and applications pending for a further seven, including a semi-automatic rifle.
Police declined to comment on the Beeld report.
"We're releasing nothing," spokeswoman Katelgo Mogale said. "Details of the incident will come out in court."