Prosecutors believe the three suspects wanted to implicate asylum seekers in their planned attack, in a case that has shocked Germans and stirred a debate about the depth of right-wing radicalism in the country's military.
The prosecutor's office named the third suspect in the case as Maximilian T, a 27-year-old German national.
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"The accused is strongly suspected of planning a severe act of violence against the state out of a right-wing extremist conviction," it said in a statement.
Former president Joachim Gauck and Justice Minister Heiko Maas were on a list of possible targets prepared by the suspects, who wanted to make their attack look like the work of militants, the prosecutor's office added.
The first soldier and the student, identified respectively as Franco A and Mathias F, were detained on April 26.
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Maximilian T was initially a witness in the case, but became a suspect after police searched several residences in Germany and France on April 26, broadcaster Hessischer Rundfunk reported.
Der Spiegel magazine, citing seized materials that belonged to Franco A, included specific details for possible attacks, including the address of the Justice Ministry and a sketch of a room at the non-profit Amadeu-Antonio Foundation in Berlin.
Investigators also found a list of different weapons and their prices, it said. German military intelligence had investigated Maximilian T in September 2015 after a fellow soldier reported that he had tried to recruit him to attack non-Germans, according to sources familiar with the matter.
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The investigation was later dropped due to a lack of evidence, Der Spiegel reported.
Chancellor Angela Merkel defended fellow conservative Defence Minister Ursula von der Leyen's handling of the case on Tuesday against critics who said the minister had wrongly implied that most soldiers were right-wing radicals.
"This is an unbelievable story with a right-wing extremist background and it was right that the defence minister did not try to minimize it but rather called it what it is," Merkel told broadcaster WDR.
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"We have to investigate whether such a thing occurs often."
Von der Leyen has apologised for her sweeping criticism of ‘weak leadership’ in the military, saying she regretted not balancing her remarks with appreciation for the work done by the majority of the armed forces.
Rainer Arnold, a Social Democrat, told the Neue Osnabruecker Zeitung newspaper that the parliamentary defence committee would demand swift reforms during a committee meeting on Wednesday.
He also criticised von der Leyen for her suggestion that more cases of extremism would likely emerge.
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"If she has knowledge that right-wing extremism is a mass phenomenon in the military, then she needs to present it publicly. Otherwise it's unacceptable conjecture," he said.
The scandal has also raised questions about the competence of staff at shelters where migrants first register before they can apply for asylum.
Franco A, who served with an army battalion stationed in France, had used a fake identity to register as a Syrian refugee and moved into a shelter for migrants in Bavaria even though he speaks no Arabic.
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The soldier had previously been detained in late January by Austrian authorities on suspicion of having hidden an illegal gun in a bathroom at Vienna's main airport, Schwechat.
Von der Leyen said Franco A had intended to have the finger prints of a Syrian asylum seeker on the gun, which he wanted to use in a possible attack and leave at the scene.
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