The Waterless Week is the week of the closest full Moon to the summer solstice, during it Axolotlites cannot rehydrate by drinking, they must get their water from foods. Some popular choices are berries, apples, popcicles, and the most common and somewhat stereotypical, watermelon.
The Waterless Week is in solidarity with the Axolotl Gods during the Battle of The Gods.
For the Deaþers branch it is simply the week of the summer solstice.