did you read the articles in the op?
They declined to comment on a possible motive, saying they would offer further information to the media about the arrest and the investigation's progress in the coming days.
However the Frankfurter Allgemeine Zeitung daily reported that the suspect "comes from the far-right scene", without providing further details.
The Bild newspaper, citing unnamed investigators, said the suspect "could belong to the right-wing extremist scene".
Lübcke was shot in the head at close range on the terrace of his home in Kassel, around 160 kilometres (100 miles) northeast of Frankfurt.
Investigators say it is unclear why the 65-year-old was killed, but a possible political motive has not been ruled out, given he had previously received numerous death threats.
Lübcke, the head of the city administration in Kassel, had spoken out in defence of migrants at the height of Europe's refugee crisis in 2015, drawing the fury of the far right.
Since his death, hundreds of posts from social media accounts tied to right-wing extremists hailed his murder.