Newspaper headlines have many atypical, telegraphic conventions in English. This particular construction is basically just a way for the journalist to highlight the most important information by putting it first in the title.
The regular straight formulation would always be "US intelligence says [that] 'Havana syndrome' [was] not caused by [a] foreign adversary".
(In the old days, journalists were taught to write their story as an "upside down pyramid". The editor was expected to cut off a suitable amount from the end of the article to make it fit, and so the meat was supposed to go first, with less and less important and relevant information trickling down to the end, where it could be mercilessly removed without seriously hampering the general narrative.)
The unabbreviated version also explains, in a way, why "adversary" is singular. The conclusion is not that some foreign adversaries were not behind it (which would be rather vague), but rather the stronger statement that they rule out that any foreign adversary was involved.