A man remains in police custody following the murder of Labour MP Jo Cox, as police revealed a link to right-wing extremism is a priority line of inquiry.
West Yorkshire Police said they are working with the North East Counter Terrorism Unit as it emerged that David Cameron was previously contacted by an MP over the safety of female parliamentarians.
Mother of two Mrs Cox, 41, 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, having been cleared by medical professionals as fit to be questioned, continues to be quizzed by detectives.
Among "numerous" lines of inquiry, the force said it is also looking at the suspect's mental health in a bid to establish a motive for the attack.
Temporary Chief Constable Dee Collins said: "A murder investigation is under way by West Yorkshire Police who are working together with the North East Counter Terrorism Unit, who will bring specialist assets in support of the inquiry.
"We are aware of the speculation within the media in respect of the suspect's link to mental health services and this is a clear line of inquiry which we are pursuing.
"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 inquiry which will help us establish the motive for the attack on Jo."
Ms Collins said it is believed the attack was "isolated, but targeted" and the person responsible is thought to have been acting alone.
The force said part of its investigation will be to establish how he was able to carry an unlawfully held firearm.
Ms Collins said a 77-year-old man remains in a stable condition in hospital after he was injured when he "bravely intervened" in an effort to help Mrs Cox.
A witness 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 Mair did not speak much about politics.
The owner of Birstall Wellbeing Centre told the Telegraph Mair visited the evening before the killing "looking for alternative therapies for his depression".
Rebecca Walker said she asked him to come back the next day.
She said: "He appeared to be quite a troubled man, didn't say very much to anyone while he was there."
Meanwhile it has emerged the MP's last words as she lay bleeding in the street were "my pain is too much".
The father of Mrs Cox's assistant Fazila Aswat has described how his daughter tried to comfort her after the attack, which left her bleeding heavily.
"She tried to help her, she tried to hit (the attacker) with her handbag but he tried to go at her. People came so he followed them and he came back again and shot her again twice," former Labour councillor Ghulam Maniyar told ITV News.
"She said her injury was so bad, and she was in her arms. There was lots of blood. She said, 'Jo, get up', but she said, 'No, my pain is too much, Fazila'. And I think those were the last words Jo spoke. She could not do anything else. She tried to comfort her."
Vigils were held across the UK on Friday evening as members of the public and politicians came together to lay flowers, light candles and stand in silence in memory of Mrs Cox.
They followed a joint visit to her home town by David Cameron and Jeremy Corbyn, where the Prime Minister issued a plea for tolerance in British politics.
The Prime Minister said the whole nation was "rightly shocked" at Mrs Cox's death, and called for people to "value, and see as precious, the democracy we have on these islands". Politics is about public service and MPs want to "make the world a better place", he said.
Across the market square from where they stood, police tape cordoned off the spot where the former aid worker was killed in what Labour leader Mr Corbyn described as "an attack on democracy".
He said Mrs Cox was "an exceptional, wonderful, very talented woman, taken from us in her early 40s when she had so much to give and so much of her life ahead of her".
Downing Street confirmed that a female MP wrote to Mr Cameron last year raising concerns about the safety of her colleagues and attacks on her personally.
A statement from Number 10 said: "The Prime Minister replied to the letter and voiced deep concern about the attacks she had suffered. The Prime Minister added that he would raise the issue with the Home Secretary. The Prime Minister also spoke to the MP about her concerns.
"The Home Secretary wrote to the MP as well and voiced her 'deep concern about the appalling incidents'. The Home Secretary also wrote to the chief constable of the MP's local police force and urged the police to 'do everything in their power' to deal with the incidents. The Home Secretary also met the MP at the end of last year."
In January, a new security package for MPs was unveiled with additional funding. Downing Street said this covered security measures at MPs' homes - in London and their constituencies - as well as constituency offices.
More than £200,000 was raised on Friday evening on a fundraising page set up by friends of Mrs Cox to support three charities "closest to her heart".
President Barack Obama phoned Mrs Cox's husband from Air Force One and offered his condolences.
"The president noted that the world is a better place because of her selfless service to others, and that there can be no justification for this heinous crime, which robbed a family, a community, and a nation of a dedicated wife, mother and public servant," a White House statement said.
The Remain and Vote Leave sides have suspended national campaigning in light of the death of Mrs Cox, who entered Parliament as MP for Batley and Spen in last year's general election.
Conservatives and Liberal Democrats have announced that they will not contest the by-election resulting from her death, giving Labour a probable free run at retaining the Westminster seat which she won with a majority of 6,057.
In an apparent reference to the EU referendum campaign, German chancellor Angela Merkel urged British politicians to "draw limits" around the language used in political debate, warning that otherwise "radicalisation will become unstoppable".
The National Police Chiefs' Council said forces are contacting MPs around the country to give security advice.
Defiant MPs have vowed to go ahead with constituency surgeries after the horrific murder.