Chancellor Angela Merkel said on Monday she had a duty to protect the right to demonstrate in Germany, regardless of the issue, and offered federal security support after an anti-Islam march was cancelled because of a terrorist threat.
Police in the eastern city of Dresden banned all outdoor public gatherings on Monday, including one by the "Patriotic Europeans Against the Islamisation of the West" (PEGIDA), a group that attracted 25,000 people to its rally last week.
The weekly PEGIDA demonstrations began last October as a local protest against the building of new shelters for refugees, and have been growing in size. Prominent members of a rising right-wing party, Alternative for Germany (AfD), are considering associating the AfD with PEGIDA.
However, counter-marches have taken place across Germany, with far larger numbers, and Merkel has condemned the group in unusually strong language as racists "with hatred in their hearts".
But at a press conference on Monday, she defended the right to protest: "Such a precious principle has to be protected. That is why everything will be done ... to ensure that the freedom to demonstrate is secured everywhere in Germany."
She offered federal help, if requested, to the regional state authorities, who are responsible for policing.
PEGIDA leaders said on Monday they would not allow themselves to be muzzled and would plan a rally for next week in conjunction with security officials.
"The threat wasn't just against me, it was also an abstract threat to the demonstration, and we have a duty to protect people," said PEGIDA leader Lutz Bachmann.
Security authorities said last Friday they had specific warnings of a risk of militant attacks on central railway stations in Berlin and Dresden.
Local police, citing information from Germany's BKA federal crime bureau, then said assassins had been called up to mingle among the PEGIDA protesters and murder one of the leaders.
PEGIDA leaders deny they are racist and say they distinguish between the secular majority among Germany's 4 million Muslims and those trying to spread Muslim values.
Justice Minister Heiko Maas, who has called the marches a disgrace for Germany, said in a statement: "Our democracy can cope with PEGIDA. The vast majority in Germany reject them and have taken to the streets in recent weeks to protest against them. This must continue to be possible."