In this context Sankara states that Badarayana reiterates - तेषां न पुनरावृत्तिः ("They no more return to this world") (Brihadaranyaka Upanishad VI.ii.15) or न च पुनरावर्तते ("He does not return again") (Chandogya Upanishad VIII.xv.1), and adds that non-return stands as an accomplished fact for those from whom the darkness of ignorance has been completely removed as a result of their full illumination and who therefore cling to that liberation as their highest goal which exists ever as an already established fact; the non-return of those who take refuge in the qualified Brahman becomes a fact only because they too have that unconditioned Brahman as their ultimate resort.
Ramachandra Dattatrya Ranade in his book, A Constructive Survey of Upanishadic Philosophy finds the concept of Transmigration of Souls developing in the Rig Veda.
(Rig Veda I.164.4), that this breathing, speedful, moving life-principle is firmly established inside these tenements (Rig Veda I.164.30), that the immortal principle, conjoined with the mortal one, moves backwards and forwards by virtue of its natural power; these two elements keep moving ceaselessly in opposite directions, with the result that people are able to see the one but are unable to see the other (Rig Veda I.164.38); that the Rishi tells us he himself saw (with his mind’s eye) the guardian of the body, moving unerringly by backward and forward paths, clothed in collected and diffusive splendor, only to keep returning frequently inside the mundane regions (Rig Veda I.164.31); and talks of those who come hither as those who are moving away, and those who are moving back as already returning hither (Rig Veda I.164.19)[5] Ramachandra Dattatrya Ranade clarifies that whenever there is recognized the possibility of the soul coming to inhabit a body as a god-like principle from without, wherever it is supposed that the soul could likewise part from the body as it came, wherever it is thought that the soul after parting from the body could lead a life of disembodied existence, and wherever it is supposed to return again to the earth and inhabit any form of existence whatsoever, there is a kind of undying life conceived for the soul.
This mode of departure is common for both the knower of the Saguna Brahman and the ignorant up to the beginning of their ways leading to immortality because it is something pertaining to this life.
The texts refer to different paths and give their descriptions but Badarayana tells us that they refer to and give only different particulars of the same path – अर्चिरादिना, तत्प्रथितेः (Brahma Sutras IV.iii.1)[9] Mundaka Upanishad (III.ii.6-9) states that at the supreme moment of final departure having become identified with the supreme Immortality (Brahman) they pure in mind become freed on every side.
As rivers, flowing down, become indistinguishable on reaching the sea by giving up their names and forms, so also the illumined soul, having become freed from name and form, reaches the self-effulgent Purusa that is higher than the higher Immutable; this Self is not attained by one devoid of strength, nor through delusion, nor through knowledge unassociated with monasticism, but the Self of that knower, who strives through these means, enters into the abode that is Brahman (ब्रह्मधाम).