The Lord, the best, the benefactor, taught the Vedas, of which the Mahābhārata is the fifth, to Sumantu, Jaimini, Bailva and Śuka, his own son and to Vaiśampāyana.
They (the students) recited separately [parts] of the Mahābhārata, which had been put together [by Vyāsa] (vedān adhyāpayām āsa mahābhāratapañcamān | sumantuṁ jaiminiṁ pailaṁ śukaṁ caiva svam ātmajam || prabhur variṣṭho varado vaiśaṁpāyanam eva ca | saṁhitās taiḥ pr̥thaktvena bhāratasya prakāśitāḥ || MBh 10.57.74–5)

What remains to be done, at this point, is to establish the identity of Vyāsa and Bādarāyaṇa. This is also a wide-spread idea, but Vedānta Deśika wants to establish through a motivation of this double name:

In the island mixed with (i.e., endowed with) Badarika (jujube) tress, out of Parāśara, Satyavatī (the mother of Vyāsa) begot a child, a destroyer of foes (parantapa), Bādarāyana, the imperishable.

dvīpe badarikāmiśre bādarāyaṇam acyutam |
parāśarāt satyavatī putraṃ lebhe parantapam ||

Unfortunately, however, I could not locate the source of the latter verse. Do you know it?

I am surprised to notice that I never discussed aikaśāstrya on this blog. You can, however, read about it in my forthcoming article on the volume I will edit with Philipp Maas, a preliminary version of which can be read here.