West Yorkshire Police said links to right-wing extremism and the suspect’s mental health are lines of enquiry in their investigation into the death of Labour MP Jo Cox.
Mrs Cox was shot and stabbed to death in the street outside her constituency advice surgery in Birstall.
West Yorkshire Police said they are working with the North East Counter Terrorism Unit as the probe into the killing of the 41-year-old continues.
The mother of two was attacked in the street outside her constituency surgery in Birstall, near Leeds at lunchtime on Thursday.
Tommy Mair, 52, was detained shortly after the attack and remains in police custody.
Temporary Chief Constable Dee Collins said: “We are also aware of the inference within the media of the suspect being linked to right wing extremism which is again a priority line of enquiry which will help us establish the motive for the attack on Jo.
“We are keeping an open mind and I do not wish to add to the speculation as we need to ensure that we conduct a professional investigation and do not compromise the need to bring the person responsible to justice.
A 77 year-old man who intervened to assist Mrs Cox remains in hospital with “a serious injury to his abdomen”. His condition is described as stable.
The attack on Mrs Cox is believed to have been “an isolated, but targeted” one, the force said, adding that they believe the person responsible was acting alone.
Police said the detained man has been cleared by medical staff to be held in detention and interviewed by detectives.
The force said part of its investigation will be to establish how he was able to carry an unlawfully-held firearm.
An eyewitness to the killing said he heard the attacker shout “put Britain first”.
It has emerged a Thomas Mair has been named in a newsletter produced by a right-wing organisation which has called for a return to apartheid-style government in South Africa and been linked to the Neo-Nazi organisation National Alliance (NA) dating back to 1999.
Mair’s brother Scott said he had a “history of mental illness, but he has had help” and both he and neighbours said he did not really speak about politics.
Explaining what happened outside Mrs Cox’s surgery on Thursday, police said the MP had just arrived at the library where she had a planned constituency meeting when she was attacked.
They said a 77-year-old man who “bravely intervened” to help, remains in a stable condition in hospital after being injured in the stomach.
Labour and Conservative politicians have been advised to talk to local police forces about the security measures they have in place for meetings with the public.
Emails from the whips office of both parties were sent out in the hours after the Labour MP was killed.
Dan Jarvis, Labour MP for Barnsley Central, said: “I know MPs are scared. We’ll be reviewing our security, but I’ll walk through Barnsley today like every Friday.”
His colleague, Jonathan Reynolds, MP for Stalybridge and Hyde, said: “I plan to keep to all my engagements today, including my surgery. I will ensure there is security present however.”
SNP MP Joanna Cherry said she would be holding a two-hour constituency surgery.
Holly Lynch, Labour MP for Halifax, said: “I will never be able to forget what has happened to Jo and, as we’ve all said, it’s incredibly, incredibly sad.
“I think you do have to bear in mind the safety of your staff at all times as well, who are quite often with you when you’re meeting members of the public, and so you cannot avoid taking these risks and concerns very, very seriously but to not let it stop you from being the effective MP that your constituents deserve.”
Labour former minister Yvette Cooper told the programme: “All of us will tell you it’s so important for MPs to be embedded in the community that we represent.
“If you go down to Asda and someone stops you to tell you about a problem with their neighbour or the fact their mum’s back garden is flooded, you write the details down on the back of a till receipt or something like that.
“That kind of daily discussion you have in the constituency is so incredibly important and no-one would ever want to put that at risk.”
Rachel Reeves, Labour MP for Leeds West, said her constituency office was temporarily closed.
She said: “We mustn’t let the actions of this man drive a wedge between MPs and the people we were elected to serve.
“The work of an MP in our surgeries, our work in the community must continue but I think it’s right today that as well as ceasing the campaigning in the referendum that we close our office.”