Tough question. I'm not a grammar expert and as a native speaker I rarely need to think about this. But I think that there is no particular rule when you use the Genitivobjekt and you just have to learn the expressions where it is used.

I think that they are very rarely used (and the German Wikipedia article on the Genitivobjekt says so, too). I can't think of many more uses than the examples on the Wikipedia page (<a href="https://de.wikipedia.org/wiki/Objekt_(Grammatik)#Genitivobjekt"><font color="#0066cc">https://de.wikipedia.org/wiki/Objekt_(Grammatik)#Genitivobjekt</font></a>).

Most of the uses of the Genitiv are related to a noun so they are not objects since an object is strongly connected to the verb/an attribute of the verb.

Maybe someone else can give you a better explanation but seeing that you had no replys yet, I wanted to try. :)