Atman and Anatman are described in the fifth chapter of Tejo Bindu Upanishad which is a dialogue between Sage Ribhu and his disciple Nidagha.
Ribhu Gita was a part of the epic Siva Rahasya like Bhagavat Gita was the part of the epic Mahabharata.
Initially, Ribhu Gita was taught to the sage Ribhu and Lord Kumara simultaneously by Lord Siva. Later, the sage Ribhu taught this to the Sage Nidagha.
Ribhu Gita is available in Siva Rahasya as the dialogue between the sage Ribhu and his disciple Nidagha. The context of Tejo Bindu Upanishad and Ribhu Gita are the same. Non-duality or Advaita is the message of the both.
Nidagha asked the sage Ribhu to explain the distinction between Atman and Anatman( that which is not Atman).
The sage Ribhu replied: Atman is the cause of all effects and at the same time, he is devoid of causes and effects. Atman is the pure consciousness. He is full of bliss and radiance. He is full of Nada. Nada means the sound that could be heard in deep meditation. Atman is beyond Nada and time. Atman is imperishable. He was from the time immemorial.
He is at the very present now and He will be in the never-ending future. He is beyond the barrier of time. Anatman is farther away from enlightenment. He is yet to realise the Maha Vakyas, the great aphorisms of Vedanta.
The Maha Vakyas are
Atman and Anatman are different states of Atman. If Atman could not realise its state of consciousness, bliss and radiance, it is Anatman.
Practically there is no such thing as Anatman. There is nothing except Atman. Everything is Atman. Only thing is there are different states of Atman.
Truth prevails if there is no deception. If deception prevails, no truth could be. Truth is the absence of deception or falsehood.
Auspiciousness is the absence of in-auspiciousness. In the absence of the effect, there could be no cause. If “I” become “thou”, “thou” ceases to be. “I” also won’t be there. If the object seen is the seer, there is no seeing. This is the very basis of the meditation on Self.
If you meditate on your Atman or Self, the object and the seer are the same, hence there is no sight at all. If nothing is seen there will be no external existence.
If there is no external stuff, seer also ceases to be. There is no mind at all. Everything becomes Brahman. “I’ and “thou” are Atman only. Brahman only. There is no Anatman.
This world exists only in the mind. Really it does not exist. It is the mind that is the world. If this world ceases to be, there will be no mind. If there is no mind, there is no world. If there is no mind and world, no egoism. No Brahman. No Anatman. Everything is the falsehood. Maya. Hence there is Anatman.
Brahma, Vishnu and Rudra are all falsehood. All the world are the falsehood. Which is true? Thou art true. Brahman is nothing other than “thou”.
Thou are that. Thou art that Brahman. Thou art in Brahma, Vishnu and Rudra. Thou art the Bliss. Thou art the radiance. There is no second one.
Thou art the pure consciousness. Everything else does not exist.
The fixed idea that I am the body is Maya or illusion. Ignorance makes the illusion.
Mind makes the ignorance. Mind alone the reason for all miseries. Mind alone is the bondage. The external world is the product of mind only.
What is the solution to the troubles created by the mind? The solution is already given. Mind alone is the solution. We already know where mind ceases to exist. By focusing your mind on Atman or self, you will become pure consciousness.
Reference: Sanskrit Text Reference