Tamerlan was killed during a furious getaway attempt last week after the attack that killed three people and left more than 180 injured. Dzhokhar is in serious condition with a neck wound and has been charged with crimes that could lead to the death penalty.

So far, no definitive answer on the Tsarnaevs’ motivations has been provided but the U.S. says it is coordinating closely with Russia.

Obama and Putin spoke last week by telephone and the American leader praised the “close cooperation that the United States has received from Russia on counterterrorism in the wake of the Boston attack,” according to a White House statement.

And the issue is likely to be high on the agenda Tuesday when Secretary of State John Kerry — a Boston native — meets Russian Foreign Minister Sergey Lavrov in Brussels.

Obama’s national security adviser, Tom Donilon, was also just recently in Moscow for talks on a range of bilateral problems.

On Syria, Russia backs President Bashar Assad’s regime while Washington supports the rebels trying to oust him. The countries are bickering over Putin’s crackdown on civil society groups and his order to halt any future American adoptions of Russian children. U.S. missile defense plans in Europe are also a sore point.

A modest improvement in counterterrorism cooperation is unlikely to fundamentally change these splits. But it could at least halt a slide in relations that has left few in the Obama administration still touting the benefits of its much-hyped “reset” four years ago.

Regarding Boston, the Russian government will only be too happy if the trail leads to Chechen militant groups it has blamed for far deadlier terrorist attacks in Russia over the last decade-and-a-half.

They’ve served as Putin’s primary rallying cry for crackdowns in the Russia’s restive south that the U.S. has responded to ambivalently. And any connection to a Islamist separatist group would dovetail with Russia’s oft-repeated argument that Syria’s rebels ought to be feared and that the Assad regime’s collapse would risk greater international terrorism.

Chechnya’s conflict began with a separatist war in the 1990s after the Soviet Union fell apart, morphing into an Islamist insurgency that Putin brutally suppressed a decade later. An estimated 100,000 people were killed.

The toll was high for Russians as well: apartment explosions in Moscow that prompted Russia’s second invasion in 1999; an attack on a Moscow theater three years later that left 129 hostages dead and a raid on the town of Beslan in 2004 that ended with 330 people, about half children, killed. In recent years, suicide bombings killed dozens on Moscow’s subway and at its airport. A train bombing claimed 26 lives.

The United States has backed Russia’s battle against terrorist groups, but has balked at the heavy-handed tactics that have caused widespread woe among civilians.

Human rights groups cite ongoing atrocities from assassinations to indiscriminate arrests based on ethnicity and say the Kremlin has too often lumped all separatists together under al-Qaida’s umbrella. This has complicated U.S.-Russian counterterrorism cooperation.

“Putin has been making the point for 14 years that Chechnya affects all of us,” said Fiona Hill, formerly the White House’s national intelligence officer for Russia under Obama and President George W. Bush.

But, she said, Russia seldom showed interest in threats specific to the United States. “That’s always been the frustration of our counterterrorism officials — that the Russians have always wanted us to focus on their issue.”

Boston’s aftermath offers a clear opportunity for better intelligence cooperation, especially if the U.S. determines that the FBI let one of the bombers slip through its fingers.

Both the U.S. and Russia want to know what the older Tsarnaev did in southern Russia for six months last year. Family members say his stay had nothing to do with separatist or Islamist rebels, but he was already apparently on Russian intelligence’s radar. They asked that the U.S. question him in 2011.

U.S. authorities found nothing that sparked their interest in Tsarnaev and stopped watching him.

Jim Treacy, who was the top FBI official in the U.S. embassy in Moscow from 2007-2009, said Russian queries for help were always taken seriously. But he said that the state of Russian assistance was fluid.

“On any given day you can get some very good cooperation,” he said. “The next you might find yourself totally shut out.”

The Obama administration has sought to improve anti-terror work with Russia, and last year added the Islamic Caucasus Emirate to a U.S. list of terrorist entities.

Still, progress has been uneven. An example: U.S. officials say they’ve been left completely in the dark about Russia’s claim to have foiled a major terrorist plot on next year’s Olympics in Sochi — near Russia’s troubled South.

The early response from local authorities in Chechnya hasn’t been auspicious.

“Any attempts to draw a parallel between Chechnya and the Tsarnaevs, if they are guilty, are futile,” said the region’s Kremlin-backed strongman, Razman Kadyrov, last week. “They grew up in the U.S., and their views and beliefs were formed there. The roots of the evil should be looked for in America.”

Kadyrov, who is accused by Washington of gross human rights violations and is reportedly on a classified U.S. sanctions list, struck a triumphalist tone by declaring that the whole world needed to combat terrorism. “We know this better than anybody else,” he said.

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Associated Press writers Adam Goldman and Eileen Sullivan contributed to this report.

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Follow Bradley Klapper on Twitter at http://twitter.com/bklapperAP

Follow Desmond Butler on Twitter at http://twitter.com/desmondbutler