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Detective Senior Sergeant Kallum Croudis has conceded police failed by not following up on Livingstone's admission that he had convictions in Australia.
Livingstone was convicted of arson and assault in Australia in 1988, related to earlier offences.
Livingstone admitted to police that he was convicted of arson while he was being held in custody following a breach of a protection order held by his children's mother Katharine Webb.
The court heard Livingstone had twice breached a court-ordered protection order preventing him from contacting Ms Webb.
The order was first issued on May 5, 2013, and Livingstone breached it in August 2013 by contacting Ms Webb.
He admitted the charge and, as it was his first offence, was granted police diversion.
Prosecutor Sergeant Kate Saxton, during questioning by Judge Marshall yesterday, conceded that was a mistake.
''My understanding from training in diversion is that diversion isn't available for offences involving breaching a court order,'' she said.
Sgt Saxton was the police prosecutor who handled Livingstone's second breach of the order on September 14, 2013.
Police opposed the defence's submission for a discharge without conviction in that case.
Livingstone was granted a discharge without conviction on November 15, 2013, by district court Judge Stephen Coyle.
At the time, Judge Coyle believed Livingstone had no previous convictions. However, police were aware Livingstone had some history of offending in Australia, Sgt Saxton said.
''Although we didn't have specifics, we were aware there was a criminal history in Australia,'' she said.
The extent of that offending was revealed on December 9 when Sgt Saxton received notification from Interpol that Livingstone had been convicted of two charges - one of setting fire to a dwelling house and one of assault, in Australia in 1988.
Judge Marshall asked Sgt Saxton why an adjournment in court proceedings was not sought until all information was available to police.
''I felt that the information I had would not be acceptable to the court,'' she said.
The Dunedin District Court had heard the second breach of the protection order was minor in nature but a witness impact statement read out by Anne Stevens, counsel for Ms Webb, at yesterday's inquest conflicted with that assessment.
''I'm sure this will happen again tomorrow or the next day. I'm constantly harassed. I'm no longer the calm and confident person I used to be. I cry in public,'' Ms Webb's witness impact statement said.
Ms Stevens asked Sgt Saxton if that was reflective of a minor breach.
''No - it had a very serious impact on her,'' Sgt Saxton said.
When pressed by Ms Webb's counsel Anne Stevens about whether the officers in charge of the investigation failed by not following up on the information, Det Snr Sgt Croudis conceded Dunedin police had failed.
"I can't answer on behalf of those officers [as to why they did not follow up], but it clearly wasn't done,'' he said.
Mrs Stevens asked Det Snr Sgt Croudis if police handling reflected a succession of failures.
He conceded that was true.
On May 15, 2013, Ms Webb left Livingstone after he brutally raped her over a period of five hours.
During that attack, Livingstone locked her in their room and continued to sexually attack her, despite their daughter banging on the door at times.
Mrs Stevens asked if it was possible Livingstone could have been imprisoned for rape if police handled the investigation better, he replied: "It's possible - it's probable, even''.
"It certainly could have been done better,'' Det Snr Sgt Croudis conceded.
Livingstone took gun from ex-flatmate
More details have emerged today of Livingstone's movements on the day he killed his children.
After shooting his children dead, it appears he then turned the gun on himself in the bedroom he formerly shared with his estranged wife.
Livingstone used a shotgun he had taken from former flatmate Philip Mans' house in Milton, Det Snr Sgt Croudis said today.
Mr Mans noticed a curtain was open when he returned home on the day of the shooting, but he thought nothing of it until informed by police his weapon had been used.
It was unclear which child was shot first, but Ellen died from a single shotgun wound, while three discharges were apparent from Bradley's wounds.
Earlier that day, Livingstone had had a supervised visit with the children at Barnardos in Dunedin .
It was to be the last of his six scheduled visits.
Barnardos receptionist Donna Rowe said Livingstone seemed normal during the hour-long visit, although he did inquire if there was to be another supervised visit as he had left a prior one early with illness.
"When he left his visit on January 15 there was nothing I could see that indicated what he was going to do later that night,'' she told Chief Coroner Judge Deborah Marshall.
Det Snr Sgt Croudis said before that visit Livingstone had purchased beer and cigarettes from a supermarket and filled a prescription for his antidepressants at a pharmacy.
Following his supervised visit with the children he visited BP Mornington and purchased petrol, including an additional quantity in a red plastic petrol canister.
Later that night, he took the shotgun and petrol canister to Ms Webb's Kiwi St home.
Livingstone entered Ms Webb's bedroom first, startling her and sending her fleeing from the house.
He then shot the children before becoming involved in a confrontation with neighbour Christopher Foot.
He shot at Mr Foot but missed and then it appears he turned the gun on himself.
Yesterday, the duplicity of Livingstone emerged at the inquest.
The inquest heard Livingstone deceived his psychiatrist, Dr Christopher Wisely, during their clinical contact.
Livingstone became estranged from Ms Webb after he brutally raped her in May 2013.
He attributed his actions to the antidepressant and smoking cessation drug Zyban.
Dr Wisely said he accepted Zyban was a factor in Livingstone raping his wife.
Many of the witnesses spoke about how Livingstone blamed his feelings and thoughts on the medication.
But his GP, Dr Coleen Lewis, was not convinced the drug contributed to his behaviour.
''It's not consistent with my experience and I see that it would be considered a very rare side effect to have psychotic behaviour,'' she said during her testimony.
Livingstone contacted her by telephone in August 2013, requesting Dr Lewis write a letter saying she prescribed him Zyban.
The phone call implied he was blaming the rape of his wife on the drug, she said.
She was so concerned by the implication that she contacted Dr Wisely and left a message, but he never replied.
Earlier in the day, a witness spoke of her concerns about Livingstone's behaviour towards Ms Webb and Ms Webb's neighbour, Melanie Foot.
Livingstone was "obsessive" about his estranged wife, the witness, who cannot be identified, said.
His behaviour had caused concern for the witness.
''He puts the s***s up me to be honest,'' she said. ''It was nothing he said or did, it was just his manner.''