Northern Ireland's Deputy First Minister Martin McGuinness laid out plans to resign, complicating the region's efforts to come up with a coherent approach to the U.K.'s decision to leave the European Union.
Northern Ireland is facing a possible snap election after Deputy First Minister Martin McGuinness resigned on Monday, ending a coalition government between Sinn Fein and the DUP.
The outgoing first minister, Mrs Foster, said his resignation was "not principled" and that she was "disappointed".
Mr McGuinness said the DUP were living in a "fool's paradise" if they thought they could return to government with Sinn Fein after an election if the RHI issue was not resolved.
"We now need an election to allow the people to make their own judgment on these issues democratically, at the ballot box", he said.
Sinn Fein has again denied that Martin McGuinness's resignation from Northern Ireland's power sharing executive was due to illness.
Mr McGuinness quit the post he has held since 2007, first sharing power with DUP leader Ian Paisley and subsequently with Peter Robinson and, latterly, Arlene Foster. Foster said she was willing to keep talks open with Sinn Féin but said she had been "disgracefully maligned in a most vicious manner". Whistleblowers allege that the generous incentives offered to businesses and farmers were abused and have highlighted delays in closing the scheme.
"The situation we face in Northern Ireland today is grave and the government treats it with the utmost seriousness", he told MPs.
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McGuinness, from the Sinn Féin party, stepped down over the so-called "Cash-for-Ash" scandal, criticizing the Democratic Unionist Party's handling of a renewable heating scheme.
Mr McGuinness, who has been experiencing health problems, confirmed that on Monday afternoon.
Mrs May said she wanted a swift resolution to the political crisis in Northern Ireland.
The DUP said it would submit to an inquiry but Foster would not step down. Sinn Fein's leadership could be about to change.
And QCs David Scoffield and Ronan Lavery, representing Northern Ireland, said it could undermine the Good Friday Agreement.
Mrs Foster has acknowledged that any forthcoming election will be "brutal" but leaders such as herself have to be careful not to totally alienate angry and disillusioned voters who simply want a government that works.