The headword is presumably extracted from the quotation given.
The verb katakonti/zw, literally I shoot down with a javelin, is more often used with an accusative (see kappa 584 and kappa 585), but here it takes a genitive.
The phrase 'the soul of Lysias' occurs -- in the genitive case, as here: th=s *lusi/ou yuxh=s -- in Dionysius of Halicarnassus, Lysias 11, referring to the genius of the Attic orator Lysias (lambda 858). But since Dionysius is admiring it, not shooting it down, the present quotation cannot come from there, and the verbal echo is accidental.
Instead we must look much later. Adler suggested an attribution to Symeon Metaphrastes (10th. cent.), but in fact the phrase as a whole occurs in a Byzantine hymn (Analecta hymnica graeca, Canones Decembris 13.20.7, lines 27-29): tw=| staurw=| eu)stoxw/tata th=s mataio/fronos tou= *lusi/ou yuxh=s kathko/ntisen.