Mr Kennedy told a New Hampshire crowd in March that he had "passed the biggest hurdle" - his wife greenlighting the run.
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As the co-founder of an environmental law firm, Mr Kennedy won plaudits for campaigning on issues such as clean water, including working to clean up the Hudson River in New York.
But his anti-vaccine views go back years and have provoked a strong backlash, including from his own family. In 2021 his sister, Kerry Kennedy, called him "very dangerous" on the issue.