As rockets and missiles streaked across Lebanese and Israeli skies on Sunday, the moment people across the region lived in fear of seemed as if it might have arrived: all-out war.

But very quickly, Israel and the Lebanese militia Hezbollah wrapped up their exchange, with both claiming victory and signaling that the fighting — for now, at least — was done.

That ambiguous result, however, revealed something: Neither Hezbollah nor its regional patron, Iran, has found a better way to respond to embarrassing Israeli strikes in a way that could both warn Israel off another attack, yet not provoke an even bigger war that could be devastating for them.

Iran’s response — if it comes — remains an unknown, and Tehran could still choose a course of action that regional observers have not predicted. But Hezbollah’s choice to stick to a limited attack is an option some regional experts now think may reflect plans from Iran, as it considers how to settle its own score with Israel.

“The Iranians keep dropping hints about striking a target with precision,” said Mohammed Ali Shabani, an Iran analyst and editor of an independent regional website, Amwaj.media. “Precision and proportion is now key to how we look at this.”

Just a few weeks ago, the region was — once again — in an extraordinarily precarious position, months after Israel launched its deadly Gaza war in response to the Hamas-led Oct. 7 attacks.

The latest round of Middle East brinkmanship began last month, when Israel blamed Hezbollah for a rocket that struck a soccer field and killed children in the Israeli-controlled Golan Heights. Hezbollah denied responsibility.

Then Israel launched a retributive escalation that quickly set the entire region on edge.

On July 30, Israel struck Lebanon’s capital, Beirut, to kill one of Hezbollah’s top commanders, Fuad Shukr. Hours later, an explosion killed Hamas’s top political leader, Ismail Haniyeh, in Tehran, where he was attending the inauguration of Iran’s new president.

The Haniyeh assassination, which both Hamas and Iran blame on Israel, was an extreme provocation for Iran’s leaders.

“If Israel can get away with killing Iranian allies in the middle of Tehran, there is no safe haven for Iranian leadership anywhere. That signal of weakness to opponents, at home and abroad, is intolerable for Iranian leaders,” said Ali Vaez, the Iran project director at the International Crisis Group. “Their dilemma was that there is no way that objective can be achieved at a low cost, and many ways in which it can backfire.”

Yet not responding, he said, is as much an existential threat as the risks of retaliation.

Part of what complicated any response for Iran was that it had already flexed its military muscle in response to an apparent Israeli strike in April that successfully targeted its embassy compound in Damascus, Syria. Back then, Tehran responded by firing a barrage of over 300 missiles and armed drones at Israel — but appeared to telegraph that attack well in advance, offering Israel and the United States an opportunity to prepare air defenses and down nearly everything that was fired.

For weeks, the concern among regional leaders and experts was less that Iran and Hezbollah wanted war, and more that their best option for a dramatic retaliation was deploying a coordinated regional show of force with other Iran-backed militant groups in Yemen and Iraq. Such a move could have resulted in a far less predictable outcome than intended by those who would have carried it out — such as hitting a site with a large number of civilians, which would have spurred Israel to jump further up the escalation ladder.

Hezbollah, which has been branded a terrorist group by Washington and is the most powerful militia supported by Iran, would have been critical to any such coordinated response.

Hezbollah’s move to act first and alone signals that option was likely ruled out, regional experts said. Hassan Nasrallah, the leader of Hezbollah, said in a speech after Sunday’s attack that “people can take a breath and relax.”

For Hezbollah, risking all-out war had a high political cost: With Lebanon still reeling from a devastating economic crisis and a yearslong political vacuum, it faces intense pressure from other segments of society not to drag the country deeper into crisis. And tens of thousands of Hezbollah’s own supporters in southern Lebanon have been driven from their homes by near-daily Israeli strikes.

Hezbollah’s critics mocked the response, sharing photos of Israeli firefighters extinguishing a fire at a chicken farm that was struck. “Sayyed Hassan has set a new equation for the Zionist enemy: We will respond to the killing of every Hezbollah fighter with the killing of a chicken,” one Twitter account posted, referring to Mr. Nasrallah.

Regardless of how Hezbollah’s response is assessed in Tehran, regional diplomats pointed to several recent comments by Iranian leaders, released shortly before and after Hezbollah’s strikes, that hint at an impending, but probably targeted and limited, retaliation.

Last week, when Hossein Salami, the commander in chief of the Revolutionary Guards Corps, visited pilgrims who were chanting slogans to avenge Mr. Haniyeh’s death, he replied, “You will hear good news about revenge, God willing.”

Shortly after Hezbollah’s strikes, Iran’s supreme leader, Ayatollah Ali Khamenei, told students at a gathering that a response “does not always mean taking up arms; rather, it means thinking correctly, speaking properly, understanding things accurately and striking the target with precision.”

Experts say, however, that these recent comments hint that Iran’s response will look less like what it did in April — though that cannot be ruled out — and more like a targeted attack.

Tehran’s main calculation is finding a response that does not risk pulling in the United States, whose warships have deployed around the region.

“The Iranians got cold feet,” said Gen. Kenneth F. McKenzie Jr., the former head of the Pentagon’s Central Command, which oversees Middle East operations. General McKenzie said that Iran would possibly retaliate by striking “a soft target” — one not heavily protected — such as an embassy or other facility in Europe, Africa or South America.

And Iranian leaders are also likely to delay any response as long as talks are ongoing to broker a cease-fire in the Gaza Strip, some U.S. officials said.

Some regional experts also pointed to intense diplomatic efforts over the prospect of negotiations on lifting sanctions.

“Iran is very pragmatic and of course has been wondering how to capitalize on this” effort by Western diplomats, Maha Yahya, director of the Carnegie Middle East Center in Beirut, said in an interview.

Mr. Khamenei, in comments viewed as underscoring a desire to renew talks with the West, said on Tuesday that there was “no barrier” to renewing negotiations over Iran’s nuclear program.

Mr. Shabani, the Iran analyst, said that by showing pragmatism and regularly projecting their thinking in advance, both Mr. Nasrallah and Mr. Khamenei may ironically have dulled their main edge against a nation far superior in intelligence prowess and military might.

“The problem for both Khamenei and Nasrallah is that the Israelis know they are logical now,” he said. “When you give away that you are not so unpredictable, it contributes to Israel’s escalation dominance.”

Some regional diplomats, however, acknowledge that even as Hezbollah and Iran look weaker today, there are ways in which Iran and its allies have already imposed a heavier toll on Israel.

By appearing to signal they have let the Gaza cease-fire negotiations take precedence — talks widely seen as being stymied by President Benjamin Netanyahu of Israel — they could find an opportunity to erode relations between Israel and its Western allies, diplomats warned. Like the Lebanese in the south, Israel’s own citizens, too, are unable to return home — and are growing increasingly frustrated with their government.

“For 75 years, we were the ones being displaced and the Israelis remained in their colonies,” Mr. Nasrallah told his followers last month. “Our homes were demolished, their colonies remained; our factories were burned, and their factories remained. Now, that has all changed.”

Hwaida Saad contributed reporting from Beirut and Eric Schmitt from Washington.

As Mexico’s president nears the end of his six-year term, his final mission is a sweeping redesign of the judiciary that he says is needed to fight corruption.

But in a potential turning point for Mexico’s democracy, President Andrés Manuel López Obrador is facing a backlash from critics who say the move is a power grab aimed at eroding judicial independence and expanding the sway of his political movement.

The proposed changes would shift the judiciary from an appointment-based system largely grounded in training and qualifications to one where voters elect judges and there are few requirements to run. The move could potentially force more than 5,000 judges from their jobs, from the chief justice of the Supreme Court down to those at local district courts.

Subscribe to The Times to read as many articles as you like.

When newcomers to Canada, the Italian couple had discovered along Quebec’s country roads the joys of the casse-croûtes, the food shacks that lie dormant in the frozen landscape during winter and then burst to life during the all-too-short warm months.

And so on a recent afternoon, the couple, Marta Grasso and Andrea La Monaca, sat side by side at a picnic table at one of these shacks, La Mollière, a lobster roll before him and a shrimp roll for her. A large blue sky spread out behind the casse-croûte, built on a promontory over the Gulf of St. Lawrence.

“You can taste the sea,” Ms. Grasso said. “We are from Sicily, so we are used to good, fresh seafood.”

The most famous menu item of Quebec’s casse-croûtes — the dish of French fries layered with cheese curds and gravy known as poutine — has become known far beyond this French-speaking province’s borders, with restaurants as far afield as Seoul specializing in the dish.

But what about the funny-sounding pogo? Or a pinceau, sometimes spelled pinso? And the guédille, whose etymology remains obscure, even though it’s a staple of casse-croûtes?

Ms. Grasso, who now calls Montreal home, was mystified when she first encountered a guédille — a sandwich consisting of a split-top hot-dog bun stuffed with seafood salad, meat or whatever is handy — on a trip three years ago. She was immediately hooked.

Her mother also became a fan during a visit from Italy last year. “She wanted to go eat a guédille every day,” Ms. Grasso said.

Casse-croûtes can be found everywhere across Quebec, many open year round.

But in far-flung areas of the vast province — as in the small towns along the St. Lawrence River or on the Gaspé Peninsula some 600 miles northeast of Montreal — they are typically mom-and-pop operations that open and close with the seasons. The menus, displayed on the exterior walls, offer American-style fast food but with a French Canadian reinterpretation.

“Casse-croûtes are summer,” said Nicole Boulay, a local resident who was eating poutine at La Mollière, in Cap-des-Rosiers, a village that is part of the city of Gaspé. “Winters here are really long.”

Mélanie Grandmont and Pascal Noël bought La Mollière shortly after getting married in 2011, turning what had been an ice cream shop into a full-fledge casse-croûte. They kept the establishment’s name, which referred to its old-fashioned ice cream.

“It’s not written the same way, but it’s also a reference to Molière,” Mr. Noël said of the 17th-century French playwright, adding, perhaps with an overabundance of caution of what customers can expect: “We have nothing to do with Molière.”

The couple are hardly miserly in their use of local products. The strawberries and raspberries on their sundaes came from their gardens. Mr. Noël made sure to secure a steady supply of local shrimp, the small but tasty Nordic kind, whose population has dwindled with the St. Lawrence’s warming waters.

One winter, the couple came to install a new dishwasher in the shack. But a wall of snow in front blocked the entrance.

“We came back in the spring,” Ms. Grandmont said.

La Mollière stirs back to life in May. The owners spend the next five months in a trailer behind the casse-croûte, no days off.

Each casse-croûte boasts of, it seems, its own secret sauce, and the trailer is also where they safeguard the ingredients of their highly classified poutine sauce — a mix of barbecue sauce with ketchup, the recipe for which was handed down by his octogenarian grandfather.

“I prepare it in secret every morning,” Ms. Grandmont said. “I bring over the ingredients to the restaurant and then hide them back in the trailer. The recipe’s in my head.”

Every casse-croûte purports to offer something unique, which is part of the restaurants’ charm, Mr. Noël said.

Casse-croûtes are “time travel machines” in the history of Quebec and the lives of its people, according to “Moutarde Chou,” a book on the establishments.

Quebec’s casse-croûtes first flourished amid the growing prosperity after World War II, said Gwenaëlle Reyt, an expert on the history of food in Quebec at the Université du Québec à Montréal.

“Casse-croûtes emerged with the boom in cars and tourism in Quebec,” Ms. Reyt said.

Although casse-croûtes became a Quebec institution, the influence on them of American fast food and car culture was strong. Burgers and hot dogs became indispensable items.

“The casse-croûtes offered dishes that we never made at home,” said Michel Lambert, an author of several books on the history of family cooking in Quebec who worked at a casse-croûte as a teenager in the 1950s. “That’s why they were considered exotic.”

Over the decades, dishes were reinvented and reimagined inside the modest shacks. Sometimes the transformations seemed more linguistic than culinary. Isn’t a pogo really a corn dog after all? And a guédille au homard is pretty similar to a Maine lobster roll, though it could be argued that the various mutations of the guédille have made it as Québécois as poutine.

At one point, Mr. Lambert researched the sandwich’s etymology.

“I never found the origins of the word guédille,” he said. “I don’t know whether it’s French or Indigenous. Maybe one day we’ll find a historical link to one man.”

At Chez Cathy, a casse-croûte in Rivière-au-Renard, another village in Gaspé city, you can get a pinso, a reinterpretation of the club sandwich. Instead of chicken, it contains ground beef patties.

“It’s one of our popular items,” said Mario Noël, who along with his two sons owns Chez Cathy.

The casse-croûte began offering the sandwich under its longtime previous owner, whose daughter was named Cathy.

“Now many other places offer it,” though elsewhere it was usually spelled “pinceau,” said Mr. Noël, who is not related to his namesake at La Mollière.

When Mr. Noël bought Chez Cathy in 2019, the deal included a secret sauce dating back to the 1960s, as well as the restaurant’s pinso and its top-selling guédilles.

Chez Cathy’s poutine had a rocky start a couple of decades ago.

“In the beginning, they used Kraft cheese instead of cheese curds,” said Danielle Samuel, who has worked at Chez Cathy for 41 years. She shook her head.

By noon on most days, the parking lot at Chez Cathy is filled, but not all the customers were exiting their vehicles; it’s one of the few places that still serve customers by the side of their cars.

For the past nine seasons, Nathalie Dufresne has flitted from car to car, taking orders and returning with food on trays that are hung on the windows.

“The locals stay in their cars, but the tourists get out and come order at the window,” Ms. Dufresne said.

“A car!” Ms. Samuel alerted Ms. Dufresne, spotting the day’s first customer, though it was still morning, as the waitress sprinted outside.

It was Dave Mainville’s day off from his job as an electromechanical technician, and he wanted to spoil himself by starting the day with his favorite, a poutine from Chez Cathy. He had been coming for years and so had his mother, with whom he was planning to share the dish.

“Casse-croûtes are open only a short season, so you want to come as often as you can,” Mr. Mainville said. “You know you won’t be able to get a poutine at Cathy’s in the month of December.”

The barrage of airstrikes that Russia launched against Ukraine over the past two days, with hundreds of drones and missiles, provided punishing evidence of Moscow’s enduring military might.

Yet for all that firepower, Russia is still struggling to reclaim a small patch of territory in its Kursk region that Ukraine seized earlier this month. And on Tuesday, its military faced attempts by Kyiv’s forces to break into the Belgorod region of Russia.

Precisely why Russia has so far failed to repel the biggest foreign incursion into its country since World War II appears to be not just a matter of personnel and lack of battlefield intelligence, but also of the priorities of Russia’s president, Vladimir V. Putin, according to Western officials and military experts.

“Putin has his focus on the Donbas,” said Gen. Onno Eichelsheim, the chief of the Dutch defense staff, who added that Kursk matters less to the Russian leader, for now. “He doesn’t care that much about this region at this moment.”

While caught off guard by the offensive into Kursk, Russia remains more intent on capturing Pokrovsk, a city that serves as a key logistics hub in Ukraine’s eastern Donbas region, and its leaders have been reluctant to pull troops from that front, the Western officials and military experts said.

“The aim of the Russian summer offensive is at least to take possession of Pokrovsk,” said Col. Markus Reisner, who oversees force development at Austria’s main military training academy and closely follows the war in Ukraine.

In the three weeks since the Kursk invasion, officials say that Russia’s slow but steady gains near Pokrovsk have, if anything, picked up.

As Russian troops continue to advance toward Pokrovsk, “any weakening of the Russian momentum caused by any redeployment is not discernible,” Colonel Reisner said.

Even so, Moscow has begun to respond in Kursk, recently moving in thousands of its forces and threatening to retaliate.

The Ukrainian incursion has “had a shocking effect on the Russians,” Christopher G. Cavoli, a U.S. Army general and NATO’s top military commander, said in remarks to the Council on Foreign Relations on Aug. 15.

But, he added: “That won’t persist forever. They’ll gather themselves together and react accordingly.”

Russia’s slow start in Kursk

Officials and experts said the Russian forces in Kursk had neither the numbers nor the experience to mount a quick defense when Ukraine’s troops blitzed over the border on Aug. 6. Those who fought did not have enough weapons or other equipment to counter the Ukrainians.

Intelligence provided by Western allies gave Ukraine a clearer picture of where Russian troops were located in the region, helping them to decide what could be captured without much resistance, said Nikolai Sokov, a former Russian and Soviet diplomat who is now a senior fellow at the Vienna Center for Disarmament and Non-Proliferation in Austria.

And it wasn’t initially obvious who was in charge of the Russian response. As of now, it’s the F.S.B. — Russia’s security agency and the successor to the K.G.B. — that is tasked with leading the response, not the Russian military’s general staff, which is in charge in eastern Ukraine.

“With no clear commander, the Russian forces are more reactive and remain on their back feet, which has allowed the Ukrainians to expand their bridgehead,” said Ralph F. Goff, a former senior C.I.A. official who served in Eastern Europe and the former Soviet Union.

To date, Ukrainian forces control about 100 settlements in the Kursk region and have captured nearly 600 Russian troops, Gen. Oleksandr Syrskyi, Ukraine’s top military commander, said Tuesday. Those numbers could not be independently verified.

A recent ramp-up

But Russia now appears to be planning for a longer-term border confrontation with Ukraine, according to an analysis by the defense intelligence firm Janes. Moscow’s response, the analysis concluded, has been “somewhat slow, but is nevertheless methodical and thorough.”

Russia has deployed attack helicopters to Kursk and recently increased its artillery strikes against Ukrainian troops there, said Mr. Sokov, the ex-diplomat.

On Tuesday, Russia’s Defense Ministry said that 400 Ukrainian soldiers had been killed and that 30 pieces of Ukrainian military equipment had been destroyed in Kursk over the previous 24 hours. That assertion also could not be independently verified, but General Syrskyi, the Ukrainian commander, separately acknowledged that Russia so far has deployed 30,000 troops to the region and is sending more every day.

Russia has deployed mostly reserve units and troops from areas in southern and northeastern Ukraine that are not part of Moscow’s main thrust toward Pokrovsk. American officials assess that Russia needs at least 50,000 troops to push Ukrainian forces out of Kursk.

But already, Colonel Reisner said, the Russian reinforcements have “slowed, noticeably” Ukraine’s momentum in the region. And it appears that Moscow has calculated that diverting enough resources to fully repel the incursion from a tactically insignificant area would not be the best use of its military power — especially as it compels Ukraine to expend its own assets to hold the territory it has taken.

“If you throw everything you’ve got to Kursk, then you are playing the Ukrainian game,” Mr. Sokov said.

The risks of collateral damage

With its intense bombardment of Ukraine this week, Russia has demonstrated that it has more than enough attack drones and missiles to devastate Ukrainian troops on its territory — assuming Moscow now has the intelligence to ascertain where they are.

But Mr. Sokov said Moscow may be mindful of harming its own citizens with a scorched-earth response in Kursk. “If you can, you might actually want to be a bit more selective about your targets,” he said.

There is also the specter of an accident at the Kursk nuclear power plant, located about 25 miles from the fighting. Operations at the plant are still active, even though it has no protective dome and is therefore “extremely exposed and fragile,” Rafael Mariano Grossi, the head of the United Nations’ atomic agency, said after a visit there on Tuesday.

Mr. Grossi said he was shown fragments of a drone that Russia has claimed tried to attack the plant, although he did not assign blame or responsibility. But if the nuclear reactor was hit, “the consequences could be extremely serious,” he said.

It is possible Ukraine is holding back some of its firepower in case it decides to launch a second surprise attack. Some Russian military bloggers have urged Moscow against sending major reinforcements to Kursk that could leave Russia vulnerable elsewhere, said Dara Massicot, a senior fellow at the Carnegie Endowment for International Peace in Washington.

The Russian airstrikes that rained down across Ukraine this week likely sought, in part, to divert global attention from the embarrassment of the incursion in Kursk, Colonel Reisner said.

But the Kremlin has made clear the incursion will not be without consequences.

“Such hostile actions cannot remain without an appropriate response,” Dmitry S. Peskov, the Kremlin’s spokesman told reporters on Monday. “There will definitely be a response.”

Lara Jakes reported from Rome and Eric Schmitt from Washington.

Promising a “once in a generation” rapprochement with Germany, Britain’s prime minister, Keir Starmer, on Wednesday pushed his plan to boost post-Brexit ties across Europe.

But while his visit to Berlin generated warm rhetoric and underscored a change of tone from the previous Conservative government’s approach, little detail was forthcoming on how repairing relations with European member states would bring about actual policy change.

After talks with the German chancellor, Olaf Scholz, Mr. Starmer said he expected that a new Anglo-German treaty, covering defense, technology, business and culture, would be struck by the end of the year as part of a broader reset of relations with the European Union.

Even as the prime minister smiled and said he was “delighted” to be standing alongside Mr. Scholz, he did not commit to specific policy changes and reiterated his election pledge that Britain would not rejoin the bloc’s economic structures. He said that Britain had “no plans” to agree a mobility scheme — of the type favored by Germany — that would help young Britons and their European counterparts cross the channel to work, though he did not rule out that prospect when asked later by reporters.

Britain’s 2016 referendum, in which the country voted by a narrow majority to leave the European Union, plunged it into a sustained political crisis and raised tensions with its member states. In the years that followed, fractious Brexit negotiations under the pugilistic prime minister Boris Johnson torched ties with many European capitals. His short-lived successor, Liz Truss, once pointedly refused to say whether the French president, Emmanuel Macron, was a friend or enemy.

So the first phase of Mr. Starmer’s re-engagement with European leaders is simply to be pleasant to them, said Charles Grant, director of the Center for European Reform, a research institute. “This is all about mood music, rebuilding trust, restoring confidence, being nice to people, being polite and showing that we have changed — which sounds obvious but is really important,” he said.

While the charm offensive was going “very well indeed,” Mr. Grant said that agreeing to specific changes would be much harder for Mr. Starmer.

Mr. Scholz, a fellow center-left politician, gave Mr. Starmer a cordial welcome, describing Germany and Britain as “good friends, close partners and trusted allies.” He said that the planned treaty was “grounded in a new spirit of cooperation,” and that Germany wanted “to take the hand that has reached out to us.”

But he also expressed his desire for more freedom of movement for young people, noting that “contacts between our societies, between Germans and people in the U.K., have declined massively after Brexit and the Covid-19 pandemic — we want to change that.”

Mr. Starmer, who said he wanted to “turn a corner on Brexit,” later flew to Paris, where he planned to attend the Paralympics opening ceremony and meet Mr. Macron, with whom he also has a strong rapport.

Before becoming prime minister, Mr. Starmer once campaigned to overturn Brexit, but he later abandoned that stance and ruled out a significant rethink.

Still, he hopes to improve the terms of a minimalist trade and cooperation agreement that was struck with the European Union by Mr. Johnson.

While Berlin might be able to offer Britain limited measures to improve economic cooperation, Germany’s trade policy is dictated by its membership of the European Union.

Any significant changes would have to be agreed not in Berlin, but in Brussels.

Mr. Starmer may try enlisting Mr. Scholz’s help in changing that. But in Berlin the main focus was on improving defense and security ties. Concerns about America’s commitment to Ukraine if Donald J. Trump wins the presidential election have underscored the importance to Germany of boosting defense cooperation. A key pillar of the new treaty with Britain, the German government said in a statement, would be to strengthen “the links between our defense communities, industries and armed forces.”

Britain’s push to improve economic ties with the European Union will likely have to wait until the fall, when a new team is in place at the bloc’s main executive body, the European Commission.

Mr. Starmer has already said that he wants to improve trade in food, animal and plant products that have been tied up in post-Brexit bureaucracy. In exchange, Britain would have to accept some European veterinary safety standards.

Mr. Grant said that to win trade concessions, Britain would have to engage with key European demands, including a comprehensive agreement on fishing rights, a deal to help young people move more freely across the continent, and Erasmus, a student exchange program that Britain quit as part of Brexit.

A youth mobility scheme could be politically sensitive in Britain, where legal immigration increased rapidly after Brexit. But the government’s objections to rejoining Erasmus relates to financial pressures, Mr. Grant said. Estimates suggest it could cost the country 2 billion pounds, or $2.6 billion, over seven years.

“Changing the nitty-gritty will be much more difficult because Britain is going to have to bite the bullet and do some of the things the E.U. wants it to do,” Mr. Grant said.

Senior American and Chinese officials meeting in Beijing discussed plans for a call between President Biden and China’s top leader, Xi Jinping, as well as talks between their military commanders in the region, as the two countries work to stabilize relations.

The White House said in a statement after the talks on Wednesday that the countries were “planning for a leader-level call in the coming weeks.”

Jake Sullivan, President Biden’s national security adviser, and Wang Yi, China’s top foreign policy official, met for nearly 11 hours over two days, the fifth round of negotiations in less than 18 months between the two top officials. They met at Yanqi Lake, a resort on the outskirts of Beijing, near the Great Wall.

The Biden administration has sought to smooth over tensions with China while continuing to turn up the pressure by curbing China’s access to the U.S. economy and technology. The meetings between Mr. Sullivan and Mr. Wang, following talks between Mr. Biden and Mr. Xi last November in California, are also aimed at showing that the rival powers can manage their differences.

“This is a mature and unusually candid channel. Sullivan and Wang have spent a lot of time together over the last year, and that has created some stability even as the relation has grown significantly more competitive,” said Rush Doshi, a former Biden administration official now at the Council on Foreign Relations, who participated in four earlier meetings between Mr. Sullivan and Mr. Wang.

One big question in the discussions is how or when the two leaders might talk in the coming months. They are running out of time to schedule the next meeting as Mr. Biden is preparing to leave office in January.

China said there had been discussion of a “new round of interaction between the two heads of state in the near future.”

Both sides said they had agreed to a video call between their top military commanders in the region. The United States said that such a call would be held in the “near future,” but China was more noncommittal, saying that it would take place “at an appropriate time.” Admiral Dong Jun, China’s defense minister, met with Secretary of Defense Lloyd J. Austin III in May in Singapore.

“This theater-command-level dialogue is essential for crisis prevention but has been resisted by the Chinese military,” said Danny Russel, a diplomacy and security analyst at the Asia Society Policy Institute. But while the two sides did agree on an initial call, they did not establish the ongoing line of high-level communication between the countries’ militaries that the Biden administration has sought.

Mr. Sullivan is also scheduled to meet on Thursday with General Zhang Youxia, vice chairman of China’s Central Military Commission, which oversees China’s military. That would be the first meeting between a Biden administration official and a vice chairman of the commission, which is led by Mr. Xi.

The United States has long sought more contact between the two countries’ militaries, in the hope that improved links might reduce the risk of miscommunication in case of a future crisis.

Worries about possible miscommunication or escalation during a conflict have increased as China has deployed military forces ever closer to longtime American partners in the region like Taiwan, Japan and the Philippines.

Just on Monday, a Chinese military surveillance plane breached Japanese territorial airspace for the first time, according to Japan, for unclear reasons. A Chinese foreign ministry spokesman said on Wednesday that the two countries were discussing the matter and that “Chinese military aircraft have no intention of intruding into the airspace of any country.”

China’s regional and global ambitions are one of the points of contention between the United States and China. The Chinese statement after the meetings between Mr. Sullivan and Mr. Wang said that the two sides should have a “correct understanding” of the relationship in order to peacefully coexist — phrasing that China uses to mean that the United States should not regard China as seeking to dominate the world.

Yun Sun, the director of the China program at the Stimson Center, a research group in Washington, said the Chinese statement indicated that Beijing was using Mr. Sullivan’s visit to “present an image of the U.S. ‘correcting its attitude,’” therefore enabling China to agree to stabilize relations.

The Chinese statement also emphasized Beijing’s unhappiness with American technology export controls and its stance on Taiwan, an island democracy that China claims as its territory.

The White House statement said American officials urged China to work with the United States to fight trafficking in narcotics that end up on American streets, and it raised concerns over the detention of American citizens in the country.

Brad Setser, a former economic policymaker in the Biden and Barack Obama administrations, said that the two countries appeared to have made more progress on security issues than on the many economic disputes that divide them. The United States has been pressing China to slow its heavy recent investments in new factories, contending that they would cause overcapacity. China has denied that overcapacity is a problem.

Olivia Wang contributed reporting.

Anna Wenske, 69, worked for decades at the national theater of East Germany, where she was born and still lives. “After the reunification, everything went kaput,” she said. She lost her job and her savings; it took her years of part-time work to reach a kind of equilibrium.

Now she resents what she considers the easy path offered to refugees while Germans suffer.

“Too many people exist on this planet and everyone wants to come to us,” she said in a sunny Weimar, “and we tell everyone welcome and we have nothing left for ourselves.” When it comes to Ukraine, she said, President Vladimir V. Putin of Russia lied when he said he would not invade, “but I don’t trust the United States any more than Russia.”

When her state, Thuringia, holds elections on Sunday, she says she will probably support the Alternative for Germany party. The radical right ethnonationalist party, known as the AfD, plays with Nazi-era language and its state branch has been classified by domestic intelligence as right-wing extremist.

But she is also tempted by a newer party, the Sahra Wagenknecht Alliance, or BSW, named for the former Communist who founded and leads it. Though the party abhors Nazism and supports the Constitution, it holds many of the same views as the AfD. “When I listen to Sahra, somehow she touches me,” Ms. Wenske said.

Germany is facing three critical state elections — Saxony also votes Sunday and Brandenburg on Sept. 22 — all in the former East Germany, where polls show such grievances are pushing many voters to the extremes, whether left or right. The expected results are already causing much hand-wringing in Berlin about the future of German democracy and the country’s failure to integrate east and west even 33 years after reunification.

The votes are being seen as a bellwether for the federal elections in September 2025, if the current government lasts that long. The three-party coalition of Chancellor Olaf Scholz is so paralyzed and unpopular that prominent members of one partner, the Greens, talk openly of a government out of ideas.

But more than anything, these elections are expected to show the persistent divisions in politics and social attitudes between west and east Germany, where extremist parties like the AfD have made by far their deepest inroads. Their success is changing the political debate nationwide, especially around issues like migration and support for Ukraine in its war against Russia.

After Friday’s knife attack in the western city of Solingen, where prosecutors say a Syrian refugee killed three people and wounded eight, the AfD and Ms. Wagenknecht have trumpeted their plans to control illegal migration and deport those who do not qualify for asylum, like the Syrian man.

That has pushed Mr. Scholz of the Social Democrats, which has lost many voters in the east, to vow to step up deportations and call for tougher European laws on asylum and deportation.

The issues of refugees, migration and knife crimes have added to the sense of estrangement in the east. In addition, the attitudes, experiences and language bred over 40 years of East German sovereignty, passed down generation to generation, are for many a source of identity and pride.

A Country Within a Country

Even today the east remains virtually a country within a country, akin to Sicily in Italy, perhaps, or Flanders in Belgium.

There remains a strong feeling of having been mocked and colonized by West Germany after unification. There is also deep sympathy for Russia, rooted in persistent anti-Americanism, where Washington is still seen as dominating German foreign policy and leading the country astray.

The AfD wants to pull Germany out of the West, out of the European Union and NATO. Some of its election posters show the German and Russian flags commingled, with the slogan: “The East does it!” or “The East is rising!”

The BSW says it wants to work within the European Union to improve it, and to ensure that NATO remains “a security union and not an alliance for war,” as its leading candidate, Katja Wolf, said.

Both are openly national socialist parties, stressing German interests and economic help to pensioners and the unemployed, who feel victimized by the aid to migrants. And both are deeply sympathetic to Russian interests and oppose further military aid to Ukraine.

Those positions are resonating. Including The Left party, the inheritor of the Communist Party, parties on the far right and far left are expected to win up to 65 percent of the votes on Sunday, according to the most recent polls in Thuringia, more than 50 percent in Saxony and about 45 percent in Brandenburg. The AfD leads in all three states.

While the center is shrinking, not everyone is ready to embrace the fringe. Though he shared many of the same views as Ms. Wenske, the retiree, Jens Wickmann, an elevator technician with three children, said he would not vote for the AfD.

“I don’t want Björn Höcke,” the AfD leader in Thuringia whom many consider a neo-Nazi, Mr. Wickmann, 56, said in the little town of Nohra. “It harms our country. We’re not Nazis.” Instead, he said, he will probably vote for the Christian Democratic Union, the center-right party that is second in the polls and is the main opposition party in Berlin.

Mr. Scholz’s Social Democratic Party, by contrast, is expected to get only 6 percent of votes in both Thuringia and Saxony. It is possible that none of the parties in his coalition, which also includes the business-friendly Free Democrats as well as the Greens, will make it above the 5 percent needed to get into the state parliaments.

Why the East Is the East

The through-the-looking-glass nature of politics in the east has proved endlessly vexing to the leadership in Berlin, as well as to the mainstream parties that once dominated German politics. It has as much to do with history as with any of the problems Germany faces.

The burden of the past is everywhere in East Germany, but doubled. First there were the Nazis, which East Germany officially labeled a West German phenomenon. Then came the legacy of the German Democratic Republic, the former Soviet-occupied East Germany, which lasted 40 years and self-identified as antifascist and anti-Western.

Steffen Mau, a sociologist at Humboldt University, said the polls underscored his thesis that German unification has not produced assimilation but in fact has solidified lasting differences between western and eastern Germany.

In an interview and a recent book, “Unequally United,” he argues that “it was an illusion that unification would bring together two things artificially separated, because over 40 years two very different societies have developed, with very different social structures.”

While western Germany is a middle-class society, the east, which lost many of its jobs soon after unification, is still relatively poor, with little accumulation of wealth, he said.

In the east, the Communist state and party and large state-owned enterprises prevented the growth of civil society, with little role for the churches or the trade unions.

“It all disappeared overnight, creating a vacuum, and right-wing actors like the AfD saw an opportunity, with a frustrated population that was homeless ideologically,” Mr. Mau said.

Is the East the Future?

Even today, party affiliations are much weaker and more volatile in the east, with more votes on the extremes and fewer for traditional centrist parties.

That “may be a harbinger of Germany’s future,” said Thorsten Benner, director of the Global Public Policy Institute in Berlin.

The BSW is the wild card. Formed only in January by dissidents from The Left party, it does not have the same stigma among some voters as the AfD, which the domestic intelligence service considers to have many extremists in its ranks.

While the AfD may get the most votes in these elections, it is shunned by all of the other parties, which vow not to collaborate with it.

That is likely to render BSW, which is running third in all three states, a key coalition partner, presenting difficult choices for the Christian Democrats, who need a partner party in the east but have ruled out a coalition with the AfD or The Left.

The C.D.U.’s lead candidate here, Mario Voigt, says the party is concentrating on topics like energy prices, education, health care and migration. “We want to frame the race around issues here that matter for the next five years and tell people not to fall into protest votes.”

Mr. Voigt admits that he benefits from fears about Mr. Höcke, the AfD leader in Thuringia. But his party has already toughened up its line on illegal migration and is sounding softer on future aid to Ukraine, much as the federal government is.

The only realistic response to the extremist parties is to try to keep them small, said Daniela Schwarzer, a political scientist and board member of the Bertelsmann Foundation, a nonprofit institute devoted to civic participation. “The hope is to demystify them before they get into office” and contain them to the regional level, she said.

But success even at a regional level would be a big breakthrough for either the AfD or BSW, with the potential for normalizing a more extremist politics in the months to come.

Tatiana Firsova contributed reporting.

Bombing eased across Ukraine after two nights of deadly barrages, but strikes near the front line killed six people and Russian troops pressed ahead in the east, closing in on the key city of Pokrovsk.

President Volodymyr Zelensky of Ukraine has called Moscow’s far-reaching bombing campaign this week one of the largest since the war began 30 months ago. Several people in the capital, Kyiv, said on Wednesday that they were pleased to have been given a respite after air-raid sirens and explosions shattered the pre-dawn calm on Monday and Tuesday.

The eastern region of Donetsk, which has seen some of the fiercest fighting this year, came under fire. A Russian attack killed four members of a family in the tiny community of Izmailivka, the state prosecutor’s office said on Facebook. The settlement is a few miles west of Russian lines and in the path of Moscow’s assault on Pokrovsk, a small city that is a vital transport hub for Ukrainian forces in the Donetsk region.

“The people died buried under the rubble,” the statement said. The regional military administration said that two other people were killed in another attack on a Ukrainian-held settlement close to the city of Bakhmut, which Russian forces captured more than a year ago after some of the most brutal combat since the full-scale invasion began.

The Donetsk region is one of two that make up the Donbas, and Russian forces have been pummeling it with daily barrages of missiles, drones and artillery fire. President Vladimir V. Putin of Russia has made controlling the whole of the Donbas a major aim.

The Ukrainian authorities have for months pressed civilians to evacuate as Russian forces advanced. But many people have stayed for reasons of poverty, ill health or attachment to their homes and farms.

Russian forces seized the Donetsk cities of Marinka and Avdiivka early in the year and have since then been moving slowly west toward Pokrovsk, which had a population of around 61,000 before the full-scale invasion.

Military analysts say that there is little chance that the city of Pokrovsk could fall imminently given the pace of Russia’s advance and the distance to the front line. But nearly half of the population has left the city, said Vadym Filashkin, the head of the regional military administration. And, in a further sign that the war is coming closer, Mr. Filashkin said on Wednesday that banks would remain open only until the end of the week.

Officials also announced a curfew for Pokrovsk from 3 p.m. to 11 a.m. No reason was given for the decision.

The advance has accelerated slightly in the last three weeks since Ukraine launched an incursion into the Kursk region of southwestern Russia, a significant development in the war.

Despite the peril in the east, the absence of a large-scale missile and drone attack overnight still came as a relief to some in Ukraine. Ukraine’s Air Force said on Wednesday that it had issued warnings about possible attacks, but that no missiles or exploding drones had been launched.

Julia Boiko, 46, a resident of Kyiv who works as a nanny, said the sounds of explosions earlier this week had terrified her 3-month-old kitten as they sheltered in the corridor of her apartment building.

“It was stressful for me and for him,” Ms. Boiko said.

Stanislav Kozliuk contributed reporting.