The case, pitting Obama against 26 states led by Texas that filed suit to block his 2014 immigration plan, is one of the biggest of the court's current term ending in June.
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The court is evenly divided with four liberal justices and four conservatives following the February death of conservative Antonin Scalia. That raises the possibility of a 4-4 split that would leave in place a 2015 lower-court ruling that threw out the president's executive action that bypassed the Republican-led Congress.
Obama took the action after House of Representatives Republicans killed bipartisan legislation, billed as the biggest overhaul of US immigration laws in decades and providing a path to citizenship for illegal immigrants, that was passed by the Senate in 2013.
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Obama's plan was tailored to let roughly 4 million people - those who have lived illegally in the United States at least since 2010, have no criminal record and have children who are US citizens or lawful permanent residents - get into a program that shields them from deportation and supplies work permits.
Obama's program is called Deferred Action for Parents of Americans and Lawful Permanent Residents (DAPA). Shortly before the plan was to take effect last year, a federal judge in Texas blocked it after the Republican-governed states filed suit against the Democratic president's executive action.
The New Orleans-based 5th US Circuit Court of Appeals upheld that decision in November. The Supreme Court's ruling is due by the end of June.
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Obama's action arose from frustration within the White House and the immigrant community about a lack of action in politically polarized Washington to address the status of people, mostly Hispanics, living in the United States illegally.
The court will decide the case at a time when immigration has become a contentious issue in the US presidential campaign, with leading Republican candidates calling for all of the estimated 11 million people in the country illegally to be deported.
Obama, who has seen many of his major legislative initiatives stifled by Republican lawmakers, has drawn Republican ire with his use of executive action to get around Congress on immigration policy and other matters including gun control and healthcare.
On the immigration action, the states contend Obama exceeded the powers granted to him by the Constitution by usurping the authority of Congress. The Obama administration called Obama's action mere guidance to federal immigration authorities on how to exercise discretion given by Congress on enforcing immigration laws.
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