The 33-year-old German man was arrested on Monday afternoon in the car that authorities say he had driven several hundred metres through a street in a busy central shopping area.
Two people - a 63-year-old woman and a 77-year-old man, both of them German - were killed.
Six other people were injured, two of them seriously.
The driver surrendered to police without resistance while still inside the vehicle.
The suspect is under investigation on suspicion of murder and attempted murder.
Authorities believe that it was a deliberate rampage but were still investigating his motive on Tuesday.
Police and prosecutors said in a statement that there was no evidence at present of a political or religious motive.
They said he had come to authorities' attention this year because of threats and "defamatory offences".
They said that police were deployed on April 17 because of a phone call from the suspect, following which he was admitted to a specialist hospital with his consent because of his "psychological condition".
He was treated there until Wednesday last week.
He had no previous convictions and there were no other pending cases, the statement said.
The social affairs ministry in Saxony state said that the man had posed no danger to himself or others while being treated, and there had been no medical reason to prevent him from leaving.
with DPA
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