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A global New York-based law firm has agreed to pay $4.6 million to settle a U.S. Justice Department investigation into whether its work for a Russia-aligned Viktor Yanukovych government in Ukraine violated lobbying laws.

The investigation stems from work that the firm, Skadden, Arps, Slate, Meagher & Flom, did with Paul Manafort, President Trump’s former campaign chairman. The case overlaps with the investigation of the special counsel, Robert S. Mueller III, into Russian interference in the 2016 election, The New York Times reports.

As part of the settlement, the law firm agreed to register retroactively as a foreign agent for Ukraine in addition to paying the government $4.6 million, representing the money it earned from its work in Ukraine.

The settlement between the firm and the Justice Department, which was made public on Thursday, is the latest indication that Mr. Mueller’s inquiry and related investigations are fundamentally challenging the lucrative but shadowy foreign-lobbying industry that has thrived in Washington.

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Mr. Yanukovych, a major client of Mr. Manafort and an ally of the Kremlin, had been accused of political motives in the prosecuting and jailing of one of his leading political rivals, Yulia V. Tymoshenko, on charges of abusing her position as prime minister in negotiating a natural gas deal with Russia.

Skadden Arps was hired to conduct a purportedly independent analysis of her prosecution and trial, and to advise the Yanukovych government on a potential future prosecution. The firm was paid more than $4.6 million, partly through the Ukrainian Ministry of Justice and partly through offshore accounts controlled by Mr. Manafort.

A Skadden Arps team led by Gregory B. Craig, a former White House counsel for President Barack Obama, produced a report concluding that, while the trial violated some of Ms. Tymoshenko’s rights, her conviction was supported by the evidence presented at trial. And the report found no evidence that the prosecution was politically motivated.

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Mr. Craig, who maintained deep connections to Washington’s Democratic establishment and its press corps, worked to shape the public relations strategy for the release of the report, according to a Justice Department filing released with the settlement.

The filing, which identifies Mr. Craig as “Partner 1” but does not name him, indicates that he arranged for a journalist to receive a copy of the report, then discussed the report with that journalist. The journalist, who is not named in the filing, is David E. Sanger of The New York Times, which published an article in December 2012 about the report quoting Mr. Craig.

Mr. Craig and Skadden Arps should have disclosed that activity under the Foreign Agents Registration Act, known as FARA, which covers both lobbying and public relations on behalf of foreign political interests, the Justice Department said. But Mr. Craig misled both his colleagues at Skadden Arps and officials in the Justice Department’s FARA unit about his interactions with the news media, leading the Justice Department to conclude that the firm was not obligated to register under the act, the settlement filing said.

The settlement “puts law firms on notice that they can’t hide behind their identity as lawyers. If they are doing lobbying work on behalf of foreign countries, they need to register under FARA,” said Rebecca Roiphe, a professor at New York Law School who specializes in legal ethics. “It also shows that the government will not tolerate false statements by lawyers.”