Yes, will email you a government reference containing list of Cays. What is your email address?
Also see Google map link https://www.google.com/maps/place/New+Cay/@23.719489,-76.1785342,15z/data=!3m1!4b1!4m5!3m4!1s0x8931ede4d6126247:0x9cdb4f9a282569!8m2!3d23.7174805!4d-76.1682299

Correct, the name is New Cay not Allen Cay.