The bucket is an item found in several places throughout the Newmaker Plane and Even Care that Paul uses on occasion to solve puzzles. The bucket first appears in Wavey and Randice's room, where the two pets will teleport away from the player before they can be caught, Paul places the bucket in one of the spots they switch between, catching Wavey in the process. Once Wavey is captured, it can be picked up along with the bucket, disabling the same bucket from being used again.
Another bucket is found in the Frozen House, where it seems to be useless; However, in Petscop 13, Paul discovers he can push the bucket through the door and go to the left of the house, where he discovers a Light Blue Tool, which follows his movements until he goes too far left, at which point is ascends upwards, and he deducts that he can place the bucket under tool and allow it to descend in when he goes to the right, which causes the tool to turn black (or a dark-grey) and allows it to be pushed wherever Paul desires. Paul uses the black tool contained in the bucket to destroy the Table Objects, gaining 15 collectibles but losing the tool and bucket in the process.
Paul uses the information gained from the Frozen House back in Even Care, creating a new file and using the bucket from Wavey and Randice's room to capture Roneth. Given that the bucket can only be used to pick up one pet at a time, it is unclear how the player could capture Roneth, Wavey, & Randice in the same file.