What is the ruling on the prayer of one who did not say "Ameen" in the first rak'ah but said it in the second, and is it a condition for one who did not say "Ameen" in the first rak'ah not to say it in the second?

If you omitted the 'ameen' in one rak'ah, you have omitted a Sunnah, which does not invalidate the prayer. If you pronounced it in another rak'ah, you have performed the Sunnah and will be rewarded for it. It is not necessary to omit the 'ameen' in all rak'ahs if you omitted it in one. Performing it is a Sunnah and brings greater reward, and pronouncing it in some rak'ahs is better than omitting it entirely.