Award availability on LX (and other M&M carriers) is notably poor for Star Alliance partners, as M&M airlines block availability.

Typically, you'll either find seats on LX flights closer to a week before your departure date or you'll find them well in advance, but not on the routes you want. JFK-ZRH is extremely difficult, if not impossible to get, via Star Alliance partners. I've had success with JFK-GVA, however, though the available dates are still sporadic.

Though ANA seems to have the most inventory when including its Star Alliance partners, its search tool is difficult to use and they charge YQ (on the other hand, redemption rates for J and above are much more competitive).

In my opinion, if you're going to be using Star Alliance, best to use United Airlines's award search engine (I think they have the same inventory availability as ANA). Once you find your desired routing and available seats, you can then search for those specific dates on ANA's website.

Keep in mind that it takes longer than 24 hours for points to transfer from MR to ANA, and LX award seats tend to be snatched up very quickly.