The Prophet ﷺ used to lie down on his right side, after offering two rakʿat (Sunna) of the Fajr prayer.  

كَانَ النَّبِيُّ ﷺ إِذَا صَلَّى رَكْعَتَىِ الْفَجْرِ اضْطَجَعَ عَلَى شِقِّهِ الأَيْمَنِ  

'A'isha reported that the Messenger of Allah ﷺ used to pray eleven rak'ahs at night, observing the Witr with a single rak'ah, and when he had finished them, he lay down on his right side, till the Mu'adhdhin came to him and he (the Holy Prophet) then observed two short rak'ahs (of Sunan of the dawn prayer).  

أَنَّ رَسُولَ اللَّهِ ﷺ كَانَ يُصَلِّي بِاللَّيْلِ إِحْدَى عَشْرَةَ رَكْعَةً يُوتِرُ مِنْهَا بِوَاحِدَةٍ فَإِذَا فَرَغَ مِنْهَا اضْطَجَعَ عَلَى شِقِّهِ الأَيْمَنِ حَتَّى يَأْتِيَهُ الْمُؤَذِّنُ فَيُصَلِّي رَكْعَتَيْنِ خَفِيفَتَيْنِ  

'A'isha, the wife of the Messenger of Allah ﷺ, said that between the time when the Messenger of Allah ﷺ finished the 'Isha' prayer which is called 'Atama by the people, he used to pray eleven rak'ahs, uttering the salutation at the end of every two rak'ahs, and observing the Witr with a single one. And when the Mu'adhdhin had finished the call (for the) dawn prayer and he saw the dawn clearly and the Mu'adhdhin had come to him, he stood up and prayed two short rak'ahs. Then he lay down on his right side till the Mu'adhdhin came to him for lqama. (This hadith has been narrated with the same chain of transmitters by Ibn Shihab, but in it no mention has been made of Iqama )  

عَنْ عَائِشَةَ زَوْجِ النَّبِيِّ ﷺ قَالَتْ كَانَ رَسُولُ اللَّهِ ﷺ يُصَلِّي فِيمَا بَيْنَ أَنْ يَفْرُغَ مِنْ صَلاَةِ الْعِشَاءِ وَهِيَ الَّتِي يَدْعُو النَّاسُ الْعَتَمَةَ إِلَى الْفَجْرِ إِحْدَى عَشْرَةَ رَكْعَةً يُسَلِّمُ بَيْنَ كُلِّ رَكْعَتَيْنِ وَيُوتِرُ بِوَاحِدَةٍ فَإِذَا سَكَتَ الْمُؤَذِّنُ مِنْ صَلاَةِ الْفَجْرِ وَتَبَيَّنَ لَهُ الْفَجْرُ وَجَاءَهُ الْمُؤَذِّنُ قَامَ فَرَكَعَ رَكْعَتَيْنِ خَفِيفَتَيْنِ ثُمَّ اضْطَجَعَ عَلَى شِقِّهِ الأَيْمَنِ حَتَّى يَأْتِيَهُ الْمُؤَذِّنُ لِلإِقَامَةِ وَحَدَّثَنِيهِ حَرْمَلَةُ أَخْبَرَنَا ابْنُ وَهْبٍ أَخْبَرَنِي يُونُسُ عَنِ ابْنِ شِهَابٍ بِهَذَا الإِسْنَادِ وَسَاقَ حَرْمَلَةُ الْحَدِيثَ بِمِثْلِهِ غَيْرَ أَنَّهُ لَمْ يَذْكُرْ وَتَبَيَّنَ لَهُ الْفَجْرُ وَجَاءَهُ الْمُؤَذِّنُ وَلَمْ يَذْكُرِ الإِقَامَةَ وَسَائِرُ الْحَدِيثِ بِمِثْلِ حَدِيثِ عَمْرٍو سَوَاءً