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These are the most open of all. The primary rule here is first come, first serve; so whoever battles it first has the primary rights to the legend. There are therefore several scenarios where the battler retains primary rights to the legend, even if things seem to have gone awry; other people should not attempt to battle it in this case, and doing so may be considered as theft:
They will forfeit primary rights to the legend in the following cases, meaning that whoever battles it next becomes the primary battler - note that this includes the original battler if they get to it again before anyone else can:
Please note that the rules around platforms mean that you may not prohibit any other player from obtaining legends from any platform land - this restriction can only be made around your main structures.
Incense legends are slightly different; but also much simpler. Unless the person using an incense clearly indicates otherwise, anything spawned from their incense is always something they have the primary rights to, and taking those legends will be regarded as stealing if permission was not given. Incense usage should still be carefully considered though; don't deliberately try and bait conflict by using it in a public warp and then complain at anyone who goes near your spawns.
If a group of people are given permission to share an incense, then the only time you can end a battle and remain the primary owner of the battling rights is if you got stuck with the waiting glitch - otherwise everyone invited has equal rights at all times. Anyone not invited is explicitly forbidden from trying to catch any Pokemon from the incense, but may catch wild Pokemon in the area (the levels should clearly give this away in 99% of cases)
All legends that are summoned are the property of the person who summoned them. However, most of these types of legends will despawn if a battle is ended; I think the only exception is Mewtwo. If you get stuck in battle with one of these, screenshot it and raise a ticket; we won't reimburse you for any other circumstance (unless the server crashes mid battle of course)