Is the wife obligated to urge her husband to acknowledge the son born of adultery and to communicate with the mother and son, and what is the ruling of Islamic law in this case? And is the son born of adultery considered a legitimate brother to the wife's son?

Adultery is one of the most heinous major sins and incurs Allah's wrath. A child born from adultery is not attributed to the adulterer. The husband must repent from the indecency by cutting off the relationship with that woman, along with feeling remorse, abandoning the sin, and resolving not to return to it. He should conceal his sin and increase his righteous deeds. The wife must help her husband in his repentance.