UFOs have returned to the national conversation in the past several years. That conversation has often invoked national security concerns. The recent passage of an omnibus spending package included a Senate request for an unclassified report on the issue. The language reflects palpable anxiety about potential technological developments among competitors.

Interesting article roughly summarizing a book the former head of the Soviet UFO study program wrote, think this is the first real discussion of the contents of the book in English. Great insight into how the program worked, and the article contains a massive collection of translated case file summaries for those interested in such things.

TL;DR to answer the question in the title: Not much! Which is about as much as any other country from what we can tell.