Take a look at the lyrics of テルーの唄:
人影絶えた野の道
...
虫の囁く草原
As I understand it, both 人影絶えた and 虫の囁く are relative clauses that modify 野の道 and 草原, respectively. But why don't we use the particle が here like:
人影が絶えた野の道
...
虫が囁く草原
When should I use の or omit the particle?
の can be used in place of が in relative clauses, although there are some exceptional rules. The basic rule of ga-no conversion is described in this question: How does the の work in 「日本人の知らない日本語」?
When ga-no conversion happens, が and の are almost the same in modern prose, but some sources say the use of の is gradually declining. I personally feel が sounds slightly more explicit and objective, whereas の is more "traditional" and has a subtle aesthetic quality suitable in lyrics and product titles. I remember my math teacher always encouraged the use of が because it sounded more academic and objective to him. (The difference is very subtle, and not everyone would agree with me. Please do not overthink this.)
In classical Japanese, a subject marker was not necessary, and you could directly join a subject and a predicate (e.g., 春が過ぎて = 春過ぎて). This is not found in modern formal prose, but you can still see a similar pattern in lyrics and fixed expressions. Especially in this song, the whole lyrics basically follow the haiku-like 7-5 pattern (七五調), so adding が or の to 人影絶えた would break the good rhythm of the song.
