Rama, the son of Dasharatha is said to be a member of Surya vamsa. Along with it, it is popularly believed that he belongs to Ikshvaku vamsa and Raghu vamsa.
To which vamsa does Rama belongs to? Surya or Ikshvaku or Raghu?
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According to Valmiki Ramayana, Ayodhya Kanda, Sarga 110, Rama belongs to solar dynasty (Suryavansa). And the other kings you have mentioned, i.e. Ikshvaku and Raghu also belongs to the same solar dynasty.
vivasvaan kashyapaaj jajne manur vaivastavaH smR^itaH |
sa tu prajaapatiH puurvam ikShvaakuH tu manoH sutaH || 2-110-6
"From Kashyapa, Vivasvan(sun-god) was born. manu was the son of Vivasvan. Manu for his part, was formerly the lord of creation. Ikshvaku was Manu' s son."
From the above verse it is clear that Manu is the son of Vivasvan (Sun) and Ikshvaku is Manu's son. In the later verses the whole solar dynasty was described.
bhagiirathaat kakutsthaH tu kaakutsthaa yena tu smR^itaaH |
kakutsthasya tu putro abhuud raghur yena tu raaghavaH || 2-110-27
"Of Bhagiratha was born kakutstha, from whom the Kakutsthas take their name. To Kakutsthas was born a son called Raghu, from whence spring Raghavas."
From the above verse it is clear that Raghu belongs to same solar dynasty.
shankhaNasya tu putro abhuut shuuraH shriimaan sudarshanaH |
sudarshanasya agni varNa agni varShasya shiighragaH || 2-110-30
shiighragasya maruH putro maroH putraH prashushrukaH |
prashushrukasya putro abhuud ambariiSho mahaa dyutiH || 2-110-31
ambariiShasya putro abhuun nahuShaH satya vikramaH |
nahuShasya ca naabhaagaH putraH parama dhaarmikaH || 2-110-32
ajaH ca suvrataH caiva naabhaagasya sutaau ubhau |
ajasya caiva dharma aatmaa raajaa dasharathaH sutaH || 2-110-33
The fortunate Sudarshana was the son of Shankhana. Sudarshana's son was Agnivarna; and of Agnivarna was born Shighraga. Shighraga begot Maru and Maru's son was Prashushruva from Prashushruva was born Ambarisha of that great radiance. To Ambarisha was born a son named Nahusha who was full of valour. Nahusha's son was Nabhaga of outstanding virtue. Aja and Suvrata were the two sons of Nabhaga and it was Aja who begot the virtuous King Dasartha."
And so was the Dasharatha also belongs to solar dynasty.
tasya jyeShTho asi daayaado raama ity abhivishrutaH |
tad gR^ihaaNa svakam raajyam avekShasva jagan nR^ipa || 34
"You are the eldest son of that Dasaratha, very well-known as Rama, the heir who can claim over the inheritance. O, King! Hence, take over your kingdom and look after your people there."
And so was the Rama belongs to the solar dynasty.