Perhaps your question is inartfully phrased. Do you mean: Can the grandmother sue the father for child support? If that's the question, then answer is yes. If the question is, can the grandmother BE sued for child support, then answer is no on 2 levels: (a) the child resides in her custody & (b) she has no legal obligation to support the child. In any event, for a full assessment, please schedule a free consultation with a Bronx Child Support lawyer.
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I really cannot read the grammar in your question so I'll guess you are asking whether a custodial grandparent can sue a non-custodial grandparent for child support.
The statute requires a support payment from the parent of the child to the other parent of the child , not from one grandparent to another grandparent.
As a grandparent under the circumstances you recite is not obligated to support the child, which typically is the case, and if there is no issue of the child becoming a publci charge, you do not have standing to sue, though the parent does.
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If you are the baby's mother and you want to sue the paternal grandmother because her son (your baby's father ) lives with her, the answer is no. You can sue the baby's father but if he has no income, you will get little. Good luck.