Here you see at least two such genitives:
ἐξήγ-ησις , εως, ἡ,
A. statement, narrative, “ἐ. ποιήσασθαι” Th.1.72; “ὑπέρ τινος” Plb.6.3.1.
II. explanation, interpretation, “περὶ τοὺς νόμους” Pl.Lg.631a; “ἐνυπνίων” D.S. 2.29; Ἐ. τῶν Ἐμπεδοκλέους, title of a work by Zeno Eleaticus; so in Gramm., Sch.Il.8.296.
ἀπόφα^σις (B), εως, ἡ, （ἀποφαίνω）
A. = ἀπόφανσις, sentence, decision, of an arbitrator's award, “διαίτης” D.47.45, cf. 33.21; κατά τινος, of an Amphictyonic decree, D.S.16.24; “ἀ. ἔγγραφος” OGI335.72 (Pergam.); of an emperor, PTeb.286.11 (ii A.D., pl.).
2. catalogue, inventory, “ἀ. δοῦναι” D.42.1,14.
3. = ἀπόφανσις, assertion, judgement, Arist.Rh. 1365b27, Epicur.Ep.3p.60U., Phld.Ir.p.75 W., Plb.1.14.8 (pl.), al.; “περί τινος” Plu.Comp.Sol.Publ.1, cf. Str.2.1.19; “καταληπτικὴ ἀ.” S.E. P.2.123.
b. answer, Plb.22.13.7; “πρὸς τὰ κατηγορούμενα” Id.24.2.5.
4. oracle, Jul.Ep.89.
5. = φάσμα, appearance, image, sc. τοῦ ἡλίου, Diog.Oen.8.
Just recently I have begun to wonder about the classifications of types of cases--genitive of blah, dative of such and such, accusative of what cha ma call it--the putting of every instance into one of n boxes. Sure, we can try and do that. And it is important to try and do that.
But on the other hand, each noun, each adjective, etc is going to have particular usages specified in LSJ. I haven't seen anybody prove or even claim that every such usage in LSJ falls into one of the n boxes.
So when we get a word like ἐξήγησις and we see that it takes the genitive to specify the content, that satisfies me now. If somebody can prove that there is some exhaustive list of n boxes, then maybe I'll try and verify that every time I find a usage that I am interested in in LSJ that it indeed falls into one of the n boxes.
I've been more guilty of this probably than anybody and like I say it is only recently that I have had a change of heart and I am glad you asked the question, but I am going to avoid answering it for the moment. Maybe in an hour I'll have a change of heart.
But until I am done cooking my chicken and eating it, I will be satisfied with LSJ.