Two German far-right politicians are being investigated by police over anti-Muslim messages on New Year's Eve.
Beatrix von Storch, deputy leader of the AfD party, had her Twitter account temporarily suspended on Monday after posting an inflammatory message.
She accused Cologne police of appeasing "barbaric, gang-raping Muslim hordes of men" after they tweeted a new-year message in Arabic.
Another politician is being investigated after supporting her.
Alice Weidel, one of the party's leaders, wrote on Facebook that authorities were submitting to "imported, marauding, groping, abusive, knife-stabbing migrant mobs".
Police say both women may be guilty of incitement to hatred. If so, they could face fines or a prison sentence.
Twitter and Facebook have deleted the posts.
AfD leaders called the actions censorship, saying the German authorities were acting like the Stasi in communist East Germany.
A new German law, which came fully into force on January 1st, means social media companies have to act against hate speech. Failure to do so could lead to fines of up to €5m ($6m; £4.4m)
Cologne prosecutors must now decide whether to launch an official inquiry.
Ms Storch's remarks came after the city's police force tweeted New Year greetings in a number of languages, including English and French as well as German and Arabic.
Twitter suspended Ms Storch's account for 12 hours in response to her post, saying it had breached the site's rules.
She then re-posted the same message on Facebook, where it was also blocked for reasons of incitement.
Cologne has been at the centre of a controversy over New Year's Eve celebrations since two years ago, when a large number of assaults against women - allegedly by men from migrant backgrounds - marred the festivities.
The following year, police in the city came under criticism for questioning hundreds of men of North African descent.
On New Year's Eve 2017, a special "safety zone" for women was set up in Berlin for the first time.