WASHINGTON (Reuters) - The Supreme Court on Monday upheld the main provision of Arizona's crackdown on illegal immigrants but threw out three other parts, handing partial victories to President Barack Obama in his challenge to the law and to the measure's conservative supporters.

In an important test of whether federal or state governments have the power to enforce immigration laws, the top U.S. court unanimously upheld the statute's most controversial aspect, a requirement that police officers check the immigration status of people they stop, even for minor offenses such as jay-walking.

But in a split ruling, the court also struck down other provisions of the southwestern U.S. state's 2010 law, the first of its kind in the country, that the Obama administration had challenged in court. The votes on those provisions were 5-3 or 6-2, with the more conservative justices in dissent.

These three provisions required immigrants to carry immigration papers at all times, banned illegal immigrants from soliciting work in public places, and allowed police arrests of immigrants without warrants if officers believed they committed crimes that would make them deportable.

Critics have argued that the law could lead to illegal racial or ethnic profiling of Hispanics in Arizona. Hispanics are the largest U.S. minority group, representing 16 percent of the population. Most U.S. illegal immigrants are Hispanics.

In the court's majority opinion, Justice Anthony Kennedy left open the possibility that constitutional or other challenges to the law, including claims it will lead to profiling, can proceed once it takes effect.

"I am pleased that the Supreme Court has struck down key provisions of Arizona's immigration law," Obama said in a statement. His administration had argued in court that the federal government has sole power over immigration, not states.

The court still dealt a setback to Obama by preserving the key part of the law months before he seeks re-election on November 6 against Republican challenger Mitt Romney, who has taken a tough stance against illegal immigration. But it was less of a setback than had been envisioned after oral arguments before the justices in April.

Conservatives also took heart. Arizona Republican Governor Jan Brewer, who has championed the measure, called the decision to preserve the heart of the statute "a victory for the rule of law" and said her state will implement it fairly and without resorting to racial profiling.

The immigration dispute was viewed as the second most important case in the Supreme Court's 2011-12 term, behind only the historic legal battle over Obama's healthcare overhaul law. A ruling in that case is expected on Thursday.

CONCERN FROM MEXICO

Arizona borders Mexico, where the foreign ministry issued a statement assailing the high political costs it said were attached to such laws and voicing concern for the civil rights of Mexicans living in or visiting U.S. states with such laws.

Kennedy cited safeguards and limits written into the law.

He said an immigrant is presumed to be in the United States lawfully if a valid driver's license or similar identification can be produced. If an immigrant lacks such identification, the officer then checks with the federal government on the immigrant's status.

Kennedy said the federal government has significant power to regulate immigration, pointing to how federal policy could affect trade, investment, tourism and diplomatic relations.

"Arizona may have understandable frustrations with the problems caused by illegal immigration ... but the state may not pursue policies that undermine federal law," he said.

The majority that struck down three challenged parts of the Arizona law also included Chief Justice John Roberts, as well as the more liberal justices Ruth Bader Ginsburg, Stephen Breyer and Sonia Sotomayor, the nation's first Hispanic justice.

The court's ninth member, liberal Justice Elena Kagan, did not take part in the immigration ruling - believed to be because she worked on the case in her prior job as Obama's solicitor general.

The decision went to the heart of a fierce national debate between Democrats and Republicans over the 11.5 million illegal immigrants the U.S. government estimates to be in the country.

OBAMA EXPRESSES CONCERNS

In his reaction to the ruling, Obama alluded to fears about racial profiling of Hispanics in Arizona.

"No American should ever live under a cloud of suspicion just because of what they look like," Obama added. "Going forward, we must ensure that Arizona law enforcement officials do not enforce this law in a manner that undermines the civil rights of Americans, as the court's decision recognizes."

In upholding the police checks, Kennedy said their mandatory nature did not interfere with the federal immigration scheme, and found unpersuasive the Obama administration argument that federal law preempted this part of the law at this stage.

He said it was improper to block that provision before state courts had an opportunity to review it, and without some showing that its enforcement conflicted with federal immigration law.

Romney had opposed the federal challenge to the Arizona law. "Today's decision underscores the need for a president who will lead on this critical issue and work in a bipartisan fashion to pursue a national immigration strategy," he said in a statement.

SCALIA ANGRILY DISSENTS

Justice Antonin Scalia read an angry dissent from the bench, saying he would have upheld the entire Arizona law.

It "boggles the mind" that the president might decline to enforce federal immigration law, Scalia said, referring to Obama's June 15 executive order stopping deportation for certain young people in the United States illegally.

Obama has vowed to push for comprehensive immigration legislation if re-elected. Opinion polls show Hispanic voters overwhelmingly support Obama.

Arizona became the first of half a dozen U.S. states to adopt laws to drive illegal immigrants out. About 360,000 of the country's illegal immigrants, or 3 percent, reside in Arizona. Most of the state's nearly 2 million Hispanics are in the country legally.

The majority opinion's sweeping rhetoric could cloud state efforts to try to curb illegal immigration.

After the ruling, the U.S. Department of Homeland Security rescinded the authority a joint state-federal task force had given Arizona to enforce federal immigration laws, senior administration officials said.

This means that even if Arizona police hold illegal immigrants, they cannot deport them unless they also broke a state law, said Evelyn Cruz, an immigration law professor at Arizona State University.

The Arizona governor's office issued a statement critical of the Homeland Security action, saying that 68 law enforcement entities in 24 states had such agreements but only Arizona's was eliminated on Monday.

Roberts said from the bench that the court's last day of the term will be Thursday, and that all remaining opinions are expected to be issued that day.

Last year, the court upheld a different Arizona law that penalizes businesses for hiring illegal immigrants.

The Supreme Court case is Arizona v. United States, No. 11-182.