Lineage of Rama has been extensively documented, not only in the Ramayana but also in Mahabaratha & Bhagavatha Purana etc. The Bhagavatha Purana is one of the 18 Purana, which among other things also speaks about the genealogy of various dynasties.
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While it is definitely possible to create a fictional family tree for a fictional person, it will not corroborate with the other texts, as they are only fictional.
Kurukshetra War of the Mahabharatha period (around 3100 BCE).
• Ikshvaku – the head of the lineage is mentioned in the Rig Veda verse 10.60.4.4.
• The Rishi, sages, Vishwamitra, Vasishta and Bharadwaja, who figure in the events of the Ramayana period and life of Rama, also find mention in the Rig Veda.