What is the ruling on a woman marrying another man, and they were blessed with two children, knowing that she was not divorced from her first fiancé, to whom she was legally (but unofficially) married? And is the marriage to the first fiancé still valid?

If a man does not divorce his wife, or she does not seek Khula (definition)' from him, or the marriage contract is not annulled judicially, she remains under his marital bond. Her marriage to another man is invalid, and they must be separated immediately. If the couple believes their invalid marriage to be valid, then the children are attributed to the second husband. Lineage is established by intercourse in a marriage believed to be valid, even if it is inherently invalid, based on the Prophet, peace and blessings be upon him, saying: "The child belongs to the bed, and the adulterer gets the stone."