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Zir is both the objective and the possessive form of the gender-neutral pronouns ze and zie. Hir can be used for the same purpose.
While ze/zir, ze/hir pronouns enjoy a degree of popularity some find that they don't offer a satisfactory degree of gender neutrality due the the ways in which the soft sounds at the beginning of both can get lost in speech and leave the pronouns sounding like the feminine "her."
Zir is used when the subject is ze. For example: "John is here! Ze brought zir books. I have more books that are zirs. I'll tell zir. Ze will be excited. Ze would never have bought this book for zirself."