Martin Sellner, head of the movement’s Austrian chapter, received €1,500 (£1,300) in early 2018 from a donor with the same name as the man charged with murdering 50 people at two mosques in Christchurch.
“We can now confirm that there was financial support and so a link between the New Zealand attacker and the Identitarian Movement in Austria,” said Mr Kurz, adding his government was considering steps to dissolve the far-right group.
Police searched Mr Sellner’s apartment on Monday and seized electronic devices to investigate a ”disproportionately high donation” from a person named Tarrant.
Australian white supremacist Brenton Tarrant, 28, has been charged with shooting dead Muslim worshippers during Friday prayers in Christchurch earlier this month.
In a video uploaded to YouTube, Mr Sellner said he “passively received” the donation. He added: “I’m not a member of a terrorist organisation. I have nothing to do with this man.”
He wrote on Twitter: “Sorry but I can’t ask every donor whether he has plans to commit a terror attack in a year, and if so to take his money back.”
Hansjoerg Bacher, spokesman for prosecutors in the Austrian city of Graz, said investigators had stumbled across the donation as part of an existing probe into possible financial offences involving Mr Sellner.
“Most donations were in the area of two-to-three figures, whereas this donation was in the low four-figure area,” he said. “This made it stand out, and the events in New Zealand put a face to this donation.”
The country’s interior ministry confirmed the domestic intelligence agency had searched Mr Sellner’s apartment in Vienna at the request of prosecutors.
Investigators are now looking into whether there were any “criminally significant” links between the Generation Identity leader and the New Zealand attacker.
“Any connection between the Christchurch attacker and members of the Identitarians in Austria needs to be comprehensively and ruthlessly investigated,” Mr Kurz said.
“It is important that the independent justice system can use all necessary means and resources to conduct its investigation together with the security services and expose these networks. There needs to be total clarity about all extremist activities.
“Our position on this is very clear, no kind of extremism whatsoever – whether it’s radical Islamists or right-wing extremist fanatics – has any place in our society.”
Austrian authorities said last week the Christchurch shooter had visited the country before the massacre. They declined to confirm when or whether he met any far-right activists during the trip, but security sources told The Independent Mr Tarrant had met right-wing extreme right-wing extremists in Europe two years ago.
Some of the views expressed in Mr Tarrant’s so-called manifesto, issued minutes before the massacre, are echoed by Generation Identity.
The pan-European ethno-nationalist group, who have branches across the continent, promote a conspiracy theory claiming that white people are being “replaced” by non-whites in Western nations.
Generation Identity’s UK chapter started in 2017 and has sent members to indoctrination camps in France, while carrying out incendiary publicity stunts at universities and protests.
Its real-life membership in Britain is small, but it has been amplified online by figures including anti-Muslim activist Tommy Robinson.
The Home Office blocked Mr Sellner from entering the country before a Generation Identity conference near London last year.
The leader of the far right group's UK chapter was invited onto Newsnight to discuss the New Zealand attacks in the wake of the atrocity, prompting accusations that the BBC was providing “a platform to racist hate”.
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