(nonstandard, possessive) Belonging to xe, their (singular). Gender-neutral third-person singular possessive adjective, coordinate with his and her.
This is xyr book.
(nonstandard, personal, chiefly Internet) The form of xe used after a preposition or as the object of a verb; that woman, that ship, etc; similar usage to him/her.
Give it to xyr (after preposition)
I wrote xyr an e-mail (indirect object)
I haven't met xyr (direct object)