The Supreme Court’s split decision on Arizona’s immigration law gave President Barack Obama an important legal victory Monday while upholding just enough of the statute to keep the issue alive as he pursues Latino voters in advance of the November election.

Indeed, the president’s advantage on the issue was clear given that top Republicans either declined to respond or, in the case of presumptive GOP presidential nominee Mitt Romney, issued statements that vaguely supported states rights without commenting on the specifics of the tough Arizona law.

The controversial “papers please” section of the law requiring police officers to try to ascertain the immigration status of people they suspect to be illegal immigrants was upheld, while the rest of the law adding state criminal penalties for immigration violations was gutted in a 5-3 ruling written by Justice Anthony Kennedy.

With the states constrained, the onus is squarely on Congress to fix the nation’s immigration system, but nothing beyond partisan posturing is likely on that front before November.

Democratic leaders and Obama quickly praised the bulk of the ruling while fretting that the remaining piece could lead to racial profiling. They called on Republicans to join them in supporting comprehensive immigration reform but do not expect the GOP to do so. Indeed, the past decade has seen a string of failures to advance immigration reform in Congress, even when one party or the other held the White House and both legislative chambers at the same time.

Still, the political benefits of the issue and the ruling were not lost on Democrats, as they angle to ensure a high turnout from Latino voters in November.

“The part left standing will only energize Hispanics,” a Senate Democratic leadership aide said, adding that “it could [be used to] persecute legal citizens of Hispanic descent.”

Indeed, Obama’s statement made plain that his administration will look to take the Supreme Court up on its tacit offer to reconsider the identification portion of the Arizona law if it is being used in a way that “undermines the civil rights of Americans.”

Senior administration officials said in a conference call with reporters that the Department of Homeland Security had no intention of stepping up its enforcement actions in Arizona — even sending directives to that effect to field offices — regardless of how many calls it gets from local police as a result of the law. Unless the suspect is a felon, it’s likely no one from the federal government will show up to pick him up for processing. And the administration immediately terminated seven task force agreements with Arizona law enforcement aimed at identifying illegal immigrants to deport.

Republicans continued to struggle to find their voice on an issue that has pitted their electoral goal of appealing to Hispanics against the need to satisfy base voters who support strong immigration enforcement measures.

Romney issued a brief statement asserting his belief in states’ rights and “duty” in helping to secure the border while generically attacking Obama. His campaign spokesman repeatedly declined to comment on the particulars of the Arizona law or the court decision, however.

Both Speaker John Boehner (R-Ohio) and Senate Minority Leader Mitch McConnell (R-Ky.) declined to respond directly to the decision.

A Boehner spokesman said he was reviewing the ruling, and a spokesman for McConnell pointed to comments from Sens. John McCain and Jon Kyl, who signed on to an amicus brief in favor of the state law.

The Arizona Republicans issued a statement saying they wanted to fully review the decision.

“We believe Arizonans are better served when state and federal officials work as partners to protect our citizens rather than as litigants in a courtroom,” they wrote.

It’s the second straight week that immigration has forced GOP leaders off their message that Obama has failed to turn the economy around.

Republicans last week struggled to respond to Obama’s decision to end deportations of many young illegal immigrants who came to the U.S. as children; Romney and GOP leaders still haven’t said whether they should be deported. But Romney said during the primaries that he would veto the DREAM Act, which would create a pathway to citizenship for some illegal immigrant children who go to college or join the military.

Some Republicans did speak out Monday. House Judiciary Chairman Lamar Smith (Texas) said states lost out in the court decision, and “to add insult to injury, the Obama administration followed up the case with a knockout punch that could send Arizona reeling backwards on immigration enforcement and border security.”

Rep. Steve King (Iowa) proposed legislation that would give states more rights to crack down on illegal immigration.

Without a Boehner statement on the ruling, it was not clear whether that proposal or any other similar bills would see floor time this year.

Most Republicans would much rather be talking about the economy.

In “this presidential race, immigration will be an issue, but the economy and jobs is such a major thing on people’s minds,” Senate Judiciary ranking member Chuck Grassley (Iowa) said. “You are not going to find the candidates for president speaking too much about immigration. If you’re running for office, you have to talk about the things that the people in this country want you to talk about.”

McCain was more blunt: “It’s very clear that we are losing a very large section of the Hispanic vote. I think that’s a fact.”

Senate Majority Whip Dick Durbin said he’s not sure whether Democrats will try to bring up the DREAM Act or other immigration legislation in order to maintain the spotlight.

The Illinois Democrat said the problem is the GOP-run House, which he said is not interested in bringing up any immigration measure.

“Now they realize that going into a general election campaign that there are many voters, not just Hispanic voters, but independent voters, who think that is an unreasonable position, so they are saying much less about it than they did during the Republican primary,” Durbin said.

Down the ballot, Congressional Democrats hoped the ruling would boost Latino turnout in key states, as the president’s recent DREAM Act order might. Several states with high Latino voter percentages — California, Colorado, Florida and Nevada — also feature several competitive Congressional races.

In the top-tier Senate contest in Nevada, Rep. Shelley Berkley (D) blasted her opponent, appointed Sen. Dean Heller, for wanting to bring the Arizona law “here to Nevada.” Like many Republican nominees in competitive states, Heller did not release a statement on the ruling. But his spokesman stated in an email that “states are frustrated with the federal government’s reluctance to enforce existing laws.”

But some GOP candidates attempted to use the ruling as a wedge in their upcoming primaries.

In Texas, former state Solicitor General Ted Cruz used the decision to attack his GOP opponent for Senate, Lt. Gov. David Dewhurst, for killing a different “bill that would have ended sanctuary cities.” Across the country, in Michigan, former state Sen. Nancy Cassis (R), a write-in candidate, criticized her 11th district primary opponent, teacher Kerry Bentivolio, for “favoring the elimination of border controls.”

Shira Toeplitz contributed to this report.