Final answer:
The specific name of the first animal shelter Rose calls in 'Rain Reign' is not given in the book; the story focuses on her systematic search for her lost dog, Rain.
Step-by-step explanation:
The name of the first animal shelter Rose calls in the book Rain Reign by Ann M. Martin is not explicitly stated. Throughout the novel, the protagonist Rose Howard, who has Asperger's syndrome, demonstrates a deep affection for her dog Rain and at one point in the story, Rose makes calls to various animal shelters in search of her lost pet after a storm. The narrative focuses on her systematic approach and her reliance on homonyms and prime numbers, but it does not provide specific names for the shelters she contacts.