There is an eastern part of the Catholic Church that is in communion with Rome. There is also a Byzantine Church that is not in communion with Rome, but constitutes the Orthodox Church. It is important to make a distinction here.
I first heard of the eastern branch of the Roman Catholic Church years ago in religion class. When the question came up, “Are priests allowed to marry?” the answer is no. However, in the eastern church priests are allowed to be married. The rule is similar to that for deacons, if the marriage came first it is fine, but even if widowed marriage is prohibited for the priests.