The official was found shot in the head and lying in a pool of blood outside his home in the state of Hesse on 2 June.
Mr Luebcke was a pro-immigration politician, known for supporting Angela Merkel’s policy to welcome refugees in 2015, when hundreds and thousands of people fled war in the Middle East and sought shelter in Germany.
Hundreds of people, including local CDU politicians, attended the official’s funeral in Kassel in June.
The suspect had a string of prior convictions for anti-migrant crime.
Among his convictions for violent crime, which date from the late 1980s to 2009, is one for a 1993 pipe bomb attack on a refugee shelter in Hesse.
“[He] says himself that he was a lone perpetrator, but of course that is not the end of the investigation for us,” Mr Seehofer said.
“The effort to clear up this political killing is not concluded with this.”
Detectives are also investigating threats sent to officials and institutions in the days after the suspect’s arrest.
It is unclear whether the threats are linked to Mr Luebcke’s death, but the killing is the latest in a series of attacks on German politicians.
In 2015 the independent politician Henriette Reker, who later become mayor of Cologne, survived being stabbed in the neck by a 45-year-old man identified as Frank S.
Convicted of attempted murder the following year, he was found to have been motivated by Ms Reker’s welcoming stance on refugees.
The CDU has said the far-right Alternative for Germany (AfD) party must share blame for the murder, saying it has legitimised a language of hate that encourages political violence.
The AfD has rejected the suggestion and said that its members are victims of left-wing violence.
Additional reporting by agencies
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