Bismillaah:

Aishah رضي اللّه عنها said:

When the Prophet صلي الله عليه وسلم had performed two Rak`ah before the Fajr prayer, he would lie down on his right side.

Takhreej:

{ Imam Al Bukhari in his Saheeh }

Abu Hurairah رضي اللّه عنه said:

The Prophet صلي الله عليه وسلم said :

" When any of you offers the two Rak'ahs
before the Morning prayer, he should lie down on his right side. "

Takhreej:

{ Imam Ahmad in his Musnad # 2/415 and graded as Saheeh by Imam ibn Khuzaimah, Ibn Hibban, An Nawaawi, Subhi Bin Hasan and others}

Imam An Nawaawi رحمه اللّه commented:

" The preferred view is that it is Sunnah based upon the apparent meaning of the Hadeeth.

{ Al Minhaj Sharh Saheeh Muslim 6/19 }

Imam Ibn Hazm رحمه اللّه commented:

" If he صلي الله عليه وسلم was unable to lie down on his right side, he would indicate as much, and he would not lie on his left

{ Al Muhallah 3/196 }

Shaykh Umar Al Bazmool commented:

This should only be done in the house as was the practice of the Prophet صلي الله عليه وسلم We are not aware of any of the Companions doing this lying down after the supererogatory prayer of Fajr in the masjid. In fact, some of them criticised this behavior.

{ Sharh Kitaab Al Asl }

*** And Allaah عزوجل knows best ***