By “Eastern Rite” I assume you mean “Byzantine”. There are at least a Dozen Eastern Rites and the Maronite is one of them.
The Maronite Church has always been in Communion with Rome, though not always in contact. The western Church thought the Maronites had died out until the Crusaders came to the Middle East and rescued them from the Muslims. The Maronite Church is the Church of the Apostillic See of Antioch. And the Maronite Patriarch is the Catholic Patriarch of all the East. We get our name from St Maron who died in the year 410. Our Liturgy is Aramaic and the recipient of the oldest Liturgical Traditions in the Church. A full history can be found here…
bkerke.org.lb/themaronites.html
The Liturgy is the product of two Traditions. That of Edessa and Antioch. The Anaphora of the Apostles, which the Maronite Church shares in common with the Church of Edessa, is the oldest Anaphora in the Catholic Church.
Anaphora=Eucharistic Prayer.
The Maronite Anaphora of the Twelve Apostles represents the oldest tradition of the Church of Antioch, being brought to Antioch by St Peter, therefore the Maronite Church, in its Liturgy, preserves the way of Worship of the Apostles and their earliest Disciples. St. John Chrysostom took the Eucharistic Prayers with him to Constantinople and they became the basis of the Byzantine liturgy.