Was the wife forbidden to her husband during the two-year period following his oath of prohibition regarding a specific action she committed, knowing that his intention was not divorce, and is the son born during this period considered an "illegitimate child"? And what is the legitimate solution to this matter?

Scholars have differed concerning the ruling on swearing to forbid one's wife. Some are of the view that its ruling depends on the intention of the one swearing. If his intention was to make the oath conditional on the wife performing a specific action in a particular manner, and the wife did not perform it in that way, then he does not break his oath. However, if he made the oath conditional on the action generally, or on a specific manner and the wife performed it in that manner knowingly, then he breaks his oath and is liable for the expiation of ẓihār or an oath, according to his intention. In any case, the child is attributed to the one who swore the oath, and intimacy with the wife before knowing about the breaking of the oath is not blameworthy. It must be noted that swearing by ḥarām (that which is forbidden) is impermissible; the lawful oath is swearing by Allah the Almighty.