"ANA search is far from exhaustive, it does not search all possibilities. To really search all options do a regular fare search on Orbitz and find all of the Star Alliance options. Then enter in each leg separately into ANA.
For example search for JFK-IST and then IST-TLV to go on Turkish via Istanbul, or JFK-MUC and then MUC-TLV to go on Lufthansa via Munich, etc."
Is Kayak just as good? I ask cuz they give you the option to choose alliances you want to fly with as opposed to click each airline separately.