The Nei weapons were of Esper creation, right? And I'm assuming the same can be said for the Elysdeon. So I'm going to take a guess here.
Would it be too far-fetched to say that perhaps the Nei sword in PSII is the same as Alis's? Perhaps after peace settled over Algo, Alis's Laconian Sword was made into the Nei Sword with the Esper's magic, or so to speak, and then hidden on Dezoris.
But then, that doesn't necessarily answer the question regarding the Elysdeon, does it?
So, what if we applied the above to the Elysdeon? Maybe the Elysdeon was created either from Rolf's Nei Sword, or Alis's Laocian Sword. Not saying that all three swords are the same, but it's something to consider. Though it leaves a hole, since if the Nei Sword from PSII is the Elysdeon, then why would its name be changed in PSIV, unless it was modified since then.
Okay, I'm goin' in circles here, but hopefully you get the point.
I just like to use some imagination at times like these, helps when there's loads of plot holes. Nothing's really for certain, unless there's some official information regarding this somewhere.