On the first day of Rosh Hashanah at Musaf, we recite piyyutim as part of the first three brachot of the Amidah, composed by R. Elazar HaKalir. On the second day, no such piyyutim are recited and we find no piyyutim until (leKeil Orech Din and) Unetaneh Tokef and Kedushah.
From Double AA I understand that this form of piyut is unusual for a musaf amidah. (The only other time we find them is on Yom Kippur.) So my question is, why do these piyyutim appear for the first day of Rosh Hashanah? And, if there is some reason why they are appropriate for Rosh Hashanah, why only on the first day and not the second?
(I understand that for whatever reason R. Elazar HaKalir only wrote one set of piyyutim for Rosh Hashanah, but given that he felt it was appropriate to have this form of piyyut at musaf, why did no-one else compose piyyutim for musaf of the second day, like R. Shimon HaGadol did for shacharis of the second day?)