The sons of Maltese journalist Daphne Caruana Galizia, who was murdered this week, have asked Prime Minister Joseph Muscat to take political responsibility for "failing to uphold our fundamental freedoms" and step down.

In a scathing Facebook post on Thursday, the sons, Matthew, Andrew and Paul Caruana Galizia, also refused to endorse Muscat's planned "million-euro" reward for any lead on their mother's killers.

"We are not interested in a criminal conviction only for the people in government who stood to gain from our mother's murder to turn around and say that justice has been served," they said.

"The Prime Minister asked for our endorsement. This is how he can get it: show political responsibility and resign."

Caruana Galizia, who linked Muscat to the Panama Papers scandal, was killed by a car bomb on Monday. She ran a popular blog that exposed high-level corruption, even by those at the very top.

Read more: Malta's overdue struggle against corruption, crooks and imbeciles 

Caruana Galizia's sons alleged that Muscat had worked to "cripple our mother financially and dehumanize her so brutally and effectively that she no longer felt safe walking down the street."

"Resign for watching over the birth of a society dominated by fear, mistrust, crime and corruption," the three wrote, addressing Muscat.

The slain journalist had filed a complaint to police earlier this month about threats she had received.

Read more: Six things to know about the Panama Papers

Muscat has denounced the murder and has vowed to "leave no stone unturned" in the search for the person or people behind the murder one of his most outspoken critics.

Investigators are looking at similarities with the other five car bombings that have happened in Malta over the past two years. None of the cases have been solved so far.

Journalists unite after brutal murder

Hundreds of journalists on Thursday held a rally outside the parliament in the capital Valletta in support of the slain journalist.

"This is one of the most despicable acts ever carried out in this country. Nobody deserves to die for exercising the right to speak," said Herman Grech, online editor at The Times of Malta, reading out a statement on the industry's behalf.

"The attack on one of us will not muzzle us. The attack on one of us will not stop us from fulfilling our role as a watchdog to the institutions. We will stand up to intimidation."

The journalists also filed a court case seeking to ensure that Caruana Galizia's sources are protected during the probe.

ap/ng (AP, AFP, dpa)