In Saxony, the police arrested 37 people from the right-wing extremist spectrum by Sunday evening after violent clashes broke out at train stations between participants in a demonstration by the right-wing extremist micro-party "The Third Way" and counter-demonstrators.
On the way to the venue in Zwickau, counter-demonstrators at Glauchau train station were attacked by several supporters of the "Third Way".
Stephen Locke
Correspondent for Saxony and Thuringia based in Dresden.
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Markus Wehner
Political correspondent in Berlin.
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Cell phone videos show the right-wing extremists chasing people on the platform, throwing stones at the train and threatening the demonstrators inside.
According to the police, two people suffered minor injuries and one person suffered serious injuries.
It is being investigated for bodily harm, serious breach of the peace and showing signs of unconstitutional organizations.
One of the attackers had shown the Hitler salute, another wore a belt with a swastika.
On the way back, when changing trains in Crimmitschau, four people from the right-wing extremist camp were seriously injured when they were attacked in a railway underpass by suspected left-wing extremists with striking tools.
Police are looking for six to ten suspects, including a woman, according to witnesses.
According to the police, around 250 people took part in the march in Zwickau on Sunday, while around 850 people took part in a counter-demonstration.
The police operation was the first test for Saxony's new Minister of the Interior, Armin Schuster (CDU).
"The police had the assembly situation in Zwickau under control at all times," he wrote on Twitter.
In the future, however, "more police presence must also be shown on the arrival and departure routes".
It is possible that the police presence in Zwickau, where more than 1,100 officers were deployed, ensured that participants went to other places to confront.
The willingness to use violence in the left camp worried him, said Schuster.
Nevertheless, right-wing extremism remains the most pressing problem.
In Berlin, according to the police, it was the “most peaceful May Day in decades”.
The police were satisfied with their large-scale operation, in which 6,000 police officers were active.
Federal Interior Minister Nancy Faeser (SPD) wrote on Twitter: “Professional preparation, high presence and immediate intervention in the event of violence pay off.
We continue to keep a very close eye on violent left-wing extremists.” However, bottles, stones and firecrackers were thrown at the police late on Sunday evening at the end of the “Revolutionary May Day Demo” in Kreuzberg.
The police used tear gas.
According to police estimates, 14,000 people took part in the demonstration, and 500 demonstrators from the left-wing extremist spectrum were estimated to be violent.
According to the police union (GdP), more than 50 arrests were made and around 30 emergency services were injured.
Several pro-Palestinian groups also took part in the demo - the previous weekend there had been anti-Semitic abuse at a pro-Palestinian demonstration.
At the May rally of the DGB at the Brandenburg Gate, an egg was thrown at the Governing Mayor Franziska Giffey (SPD), but it was intercepted by a security officer with an umbrella.