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"It can not be one rule for those who set them and another for the British people", Labour leader Keir Starmer said.
Johnson's office said Cummings made the journey to ensure his 4-year-old son could be properly cared for as his wife was ill with COVID-19 and there was a "high likelihood" that Cummings would himself become unwell.
Mr Shapps said at the press conference that Mr Cummings would have been thinking about finding "the best possible option" to support his son.
The spokesperson continued: "Officers made contact with the owners of that address who confirmed that the individual in question was present and was self-isolating in part of the house".
Downing Street said Cummings' "actions were in line with coronavirus guidelines".
Officers "explained to the family the arrangements around self-isolation guidelines and reiterated the appropriate advice around essential travel", the force said.
Finance chief Rishi Sunak also defended Mr Cummings, saying: "Taking care of your wife and young child is justifiable and reasonable, trying to score political points over it isn't".
But critics of the government expressed outrage that Cummings had broken stringent rules that for two months have prevented Britons from visiting elderly relatives, comforting dying friends or even attending the funerals of loved ones. He said on March 23 that people "should not be meeting family members who do not live in your home".
"It's as simple as that", Ed Davey, the acting leader of the Liberal Democrats said, according to the Guardian.
"You guys are probably all about as right about that as you were about Brexit: do you remember how right you all were about that", he added.
Speaking to the BBC, former Conservative MP David Liddington, who was de facto deputy Prime minister under Theresa May, said: "There's clearly serious questions that No 10 are going to have to address not least because the readiness of members of the public to follow government guidance more generally is going to be affected by this sort of story".
The Scottish National Party's leader in the House of Commons Ian Blackford said Cummings "must resign or be fired".
Mr Cummings tested positive for coronavirus, having developed symptoms over the weekend 28 and 29 March.
In a piece published by the Spectator on 25 April about how the magazine's authors were spending lockdown, Cummings was careful not to mention his trip north when he wrote that "at the end of March and for the first two weeks of April I was ill, so we were both shut in together".
Opposition events have known as for the prime minister to sack his chief adviser over the obvious lockdown breach, which was revealed in an investigation by the Guardian and the Mirror.
Dominic Cummings drove from his home in London to his parents' Durham farm on March 27, only a few days after Johnson initiated a national lockdown.
Alastair Campbell insisted that the scandal surrounding Dominic Cummings is proof that this government sees itself as above the law.
Earlier today, a defiant Mr Cummings responded "who cares" when he was asked by reporters whether his actions looked bad.
Professor Neil Ferguson, one of the United Kingdom government's most high-profile scientific advisers, resigned from a committee after it emerged a woman from outside his household had visited his home on two occasions.
Scotland's chief medical officer, Catherine Calderwood, resigned in April after breaking her own rules to visit her second home twice.