With respect to permission of a Wali for muta, what is the case of an umarried girl whose parents have got divorced and she stays with her mother and has broken all ties with her father and his family?


And what is the case for a such a permission when father has deserted (although not divorced) the family (for a second marriage and/or other reasons) and is no longer interested in the affairs of his family? 

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If what you mentioned applies to both cases (desertion), which means that he does not care about his daughter, and gives her the freedom to do what ever she wants to do, then for S. Sistani, she still needs to take his permission, but on mandatory precaution basis, where she can refer to another marjee taqleed who allows such marriage.
In this case, she can refer to S. Hakeem, provided that the muta contract is based on no intercourse.