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The under-fire politician made the announcements on Friday afternoon after she revealed last night she was registered to vote in an electorate that she was not living in.
The Dunedin-based MP said she would not resign over that or her benefit history - in which she failed to declare all her flatmates when she was claiming a sole-parent benefit while a law student in the 1990s - despite considering it and discussing it with co-leader James Shaw.
"I'm not resigning as co-leader or as a member of Parliament," she said.
"I will continue to stand for New Zealanders who are poor and who are treated with discrimination by the welfare system. This work is important. It is more important than one person."
However, Ms Turei said she would not seek to be part of a cabinet in a Labour-Greens government after the general election on September 23.
Ms Turei admitted this was a concession to the Labour Party, who had voiced "concerns" over the revelations. She was disappointed that she would not be a minister, but said the work she was doing on welfare was more important than her personal career.
Last night, the Dunedin-based MP admitted that she enrolled at a Mt Albert address where she did not live in 1993 so she could vote for a friend. Enrolling to vote at an address you do not live at is an offence.
Ms Turei also confirmed her mother was a flatmate for part of the period she claimed the benefit in the 1990s. She said they were financially independent at the time.
And last month she revealed that she did not disclose to Work and Income in the 1990s that she had extra flatmates while she was a solo mother on the domestic purposes benefit.
Labour leader Jacinda Ardern told reporters on Friday afternoon her party's relationship with the Greens would not be affected by Ms Turei's revelations.
Labour and the Greens have a Memorandum of Understanding which commits them to working together until election day on September 23.
Former Green MP Sue Bradford, who is a beneficiary advocate, said earlier today that Ms Turei had underestimated the political and legal risk of her admission.
She said Ms Turei had been "courageous" in announcing a radical welfare policy and telling her story, but did not appear to have calculated the "detrimental" impact it could have on her political career.
"I'd like to hope she could survive, I'd hate to see her step down as a result of this. I'm not prepared to commit either way on this, but I think she has put herself in a precarious position."
Former Winz head Christine Rankin said it was common for beneficiaries to be living with their parents, but this had to be disclosed because it would affect their welfare payments.
"Benefit fraud alone is appalling. Having your mother is just one of those things, it could happen to lots of people. But it needs to be assessed and declared.
"Because when you become a beneficiary, you literally hand over your life to Work and Income. You have to tell them everything."
Ms Turei said last night that she was the sole provider for her daughter Piupiu when she was on the domestic purposes benefit.
Her enrolment at an address she did not live at was "a mistake - one of many I, like many other people, made as a young person".
Act leader David Seymour earlier told Newstalk ZB he doesn't buy Ms Turei's explanations, accusing her of "spinning, spinning, and spinning".
He said if Ms Turei was any other MP she would have been gone by now.
Serial litigant Graham McCready this morning said he was taking a private prosecution against Ms Turei, alleging she had committed electoral fraud and filed false tax returns.
Mr McCready, who has taken legal action against several politicians, said he had made complaints to Inland Revenue and the Electoral Commission.