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If there is MileSAAver availability on aa.com, you just have to call BAEC back and be insistent. A few months ago, I found MileSAAver availability on aa.com for a four-flight itinerary. The BAEC agent initially said that none of the four flights was available. When I told her that all four were available as MileSAAvers on aa.com, she said BA can only book the award seats that AA makes available to BA. When I told her that all MileSAAver award seats are made available to BA, she did something else with her computer and -- lo and behold -- was able to book each of the four flights which I had identified.