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President Trump, his eldest son, and his former lawyer, Michael D. Cohen, were among a roster of rich and powerful people who received gun licenses from the New York Police Department in return for special favors, a former lieutenant has claimed in court papers.

The former lieutenant, Paul Dean, said the men received permits to carry guns in New York City without the proper paperwork after donating to two charities with close ties to the department. They were among a list of other well-connected people who Mr. Dean said benefited from a “systematic culture of corruption” that stretched from the department’s gun licensing division to the upper echelons of the department.

The police commissioner, James P. O’Neill, said the allegations by Mr. Dean, who has pleaded guilty to approving gun licenses in exchange for bribes while he was the second-in-command of the department’s licensing division, are “highly suspect.”

“These claims were investigated thoroughly by our I.A.B. — internal affairs — the F.B.I.’s public corruption unit and overseen by federal prosecutors,” Mr. O’Neill said on Thursday, a day after Mr. Dean made the allegations. “These investigations found no credible basis to charge any other individuals.”