Amateur video shows skinheads chasing a young man as far-right groups protest over death of man.
Germany will not tolerate "vigilante justice", Chancellor Angela Merkel's spokesman said on Monday, after hundreds of far right protesters staged a violent demonstration following a fatal stabbing that authorities blamed on a Syrian and an Iraqi.
Sunday's protest in the eastern city of Chemnitz was the latest manifestation of unrest since Merkel's government allowed about 1 million asylum seekers to enter Germany in 2015, triggering a shift to the right in German politics.
Television news channels broadcast amateur footage of skinheads chasing a foreign-looking man through the streets. Other clips showed hundreds of demonstrators shouting "We are the people!", a slogan used by far-right supporters.
Police said around 800 demonstrators had taken to the streets on Sunday, hours after one man was killed and two other people were injured in the stabbing incident overnight.
Local police were trying to avert further violence as dozens of far-right demonstrators and hundreds of leftist protesters staged separate rallies in the city on Monday evening.
"We don't tolerate such unlawful assemblies and the hounding of people who look different or have different origins, and attempts to spread hatred on the streets," Merkel's spokesman Steffen Seibert told a regular news briefing.
"That has no place in our cities and we, as the German government, condemn it in the strongest terms," he said. "Our basic message for Chemnitz and beyond is that there is no place in Germany for vigilante justice, for groups that want to spread hatred on the streets, for intolerance and for extremism."
TRT World's Ira Spitzer has been watching the events from Berlin.
Roland Woeller, interior minister of the state of Saxony, which includes Chemnitz, said authorities would not allow "anarchists" to run rampant. He urged all sides to remain calm and rely on official police information for updates, saying that "misinformation and lies" were circulating in social media.
Police said the stabbing occurred after a row, but gave no further details. Local prosecutors said on Monday they had arrested two suspects, a 22-year-old Syrian man and a 21-year-old Iraqi man. No additional details were immediately available about the two men.
Authorities have described the victim as a 35-year-old German man, and said two other German men, aged 33 and 38, were also injured. The Frankfurter Allgemeine Zeitung newspaper said the victim was German-Cuban, a father and trained carpenter.
Charges were also filed against four protesters at Sunday's rally, including two charged with bodily injury.
Some of the protesters had thrown bottles and rocks at police officers. Civilians were also attacked and harassed, including an 18-year-old Afghan man and a 15-year German girl, a police spokeswoman said.
The demonstration followed calls on social media by far-right groups for protests over the man's death.
The arrival of huge numbers of migrants from the Middle East three years ago has fueled support for far-right groups such as PEGIDA and the Alternative for Germany (AfD), now the main opposition party in parliament.
Tweeting about Sunday's incident, AfD politician Markus Frohnmaier said: "If the state is no longer to protect citizens then people take to the streets and protect themselves. It's as simple as that!"
Martina Renner, a lawmaker for the radical Left party, accused the far-right of trying to exploit a murder for its own political ends.
"A terrible murder, the background to which is still unclear, is being instrumentalised in the most repugnant way for racist riots in Chemnitz," she tweeted.
The violence in Chemnitz is likely to put further pressure on Merkel's conservatives, who last week faced accusations of ignoring the rise of far-right groups in the eastern state of Saxony, where Chemnitz lies.