Go to Travelocity (for example)

Click on Flights

Put in where you want ot go from to, and then click on my dates are flexible

Tell it to search only the airline you are interested, but remember to add in any partner ones, so for United, you will wan tot put in some other Star Alliance ones

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The fares will come up, then you have to play with the dates to see when they are available. Click on the start date you want, and then choose your end date. Then go take all that information, head over to the airline's site to get the bonus miles and see if they match the fare.