When you have Psychic Sight, once per day you can expend your psionic focus when manifesting detect psioinics to manifest arcane sight (well, its “psionic equivalent” anyway) as well. At that point, you have two separate effects on you, one detect psionics and the other arcane sight. They are independent of one another, and the arcane sight will last 1 minute/level, even if you stop concentrating on the detect psionics effect. Likewise, while you can manifest incarnate on the detect psionics and cause it to become permanent, the arcane sight effect will still last 1 minute/level. Nothing in the feat description ties the duration of the arcane sight effect to the duration of the detect psionics effect.
On the other hand, arcane sight is a valid target for permanency, so its “psionic equivalent” is presumably a valid target for incarnate. This would presumably cost the same 1,500 XP and require the same 11th level as permanency does for arcane sight. But Mind’s Eye neglects to spell this out, so you’ll have to double-check exactly what “psionic equivalent” really means; that’s not a defined term in the rules.
For that matter, it’s not really clear that this psionic arcane sight is actually being manifested as a psionic power. It just says you “gain” it. If this is, say, a psi-like or supernatural effect, then it won’t be a valid target for incarnate.