Is the money received by the children or daughter-in-law from the state in exchange for caring for their incapacitated parents permissible for them, or is it not allowed to be taken, or must it be divided equally among the children?

It appears that the one performing the service, whether it is the son or someone else, is the one entitled to the grant. If the son hires a maid for his father, he has the right to take the grant, because the maid receives her wage from him, and she should not combine the two wages. The one who directly performs the service himself, or through his hired help or agent, is entitled to the grant. Likewise, if the son's wife is the one taking care of the incapacitated person, there is no objection to her registering her name to obtain the grant, as she is entitled to it by virtue of her service.