This reminds me -- I need to gather information on whether one can run while Flying. My group is under the impression that the best movement you can perform is double-Move, but that seems strange to me, considering it is 'tactical' flight wherein you retain your Dex bonus to AC.
The only real Flight writeups I know of are in the Fly skill and the very brief paragraph in the Bestiary glossary.
Which, while brief, is still better than the Burrow writeup which doesn't exist at all.
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I believe the assumption is that you can run with any movement mode for which you have a speed. If you're swimming without a swim speed, you can't run, but if you have one you can.
Note that the movement section under "Run" never says anything about it being specific to land movement, and none of the movement mode sections contradict the ability to run. In fact, the base "movement" section never mentions anything to indicate that it's specific to land other than the name of the speeds (which I assume are such because land is the norm).
The Fly spell even has to specifically disallow running.
Basically, this falls under "It's not explicitly called out, but it's technically legal, not explicitly forbidden, hinted at via other rules and just plain makes sense."