What is the legitimate Islamic solution for this wife whose husband has abandoned her and refuses to divorce her or take her back, and what is the duty of the children towards her?

If the father has abandoned the mother and left her in a state of suspension, he is legally sinful and must either keep her honorably and fulfill her rights, or separate from her kindly. The mother is still under the husband's marital bond, so it is not permissible for her to marry or entertain marriage proposals until he divorces her and her waiting period ends, based on the Almighty's saying: "And all married women [are forbidden to you]." Efforts should be made to reconcile the parents. If this is not possible and the mother desires separation, there is no harm in helping her to remove the injustice from her. If the father does not respond to the request for divorce, she should refer her case to the competent authorities. The mother's contemplation of marriage, with the intention of preparing for it by pursuing a permissible separation from her husband, is not blameworthy.