[Gateway103]

Some ideas, by no means exclusive:

1. (Works independent of Nation/Magic Path/Research)

Use armour piercing range attack units, e.g. Crossbow, usually can get them from conquered indie provinces. These guys are slow, fire every 2 or 3 turns, but 50 or so crossbow man can take a Body Etheral Sphynx's HP down by about 100 every salvo. If your opponent doesn't have access to Body Ethereal or other protective spells, it'll be a whole lot easier.

2. (Nation + Magic Path + Research dependent)

use well equipped Assassins, if you have access to them, to get rid of the Sphynx. Playing as Pythiums before, my success rating against the Arco Sphynx has been 2 out of 3 with Assasins, despite its huge 1500 HP at capitol.

3. (Research + Magic Path dependent)

Give strong leaders fire resistance artifacts. I think Buring Pearls, among others, does the trick.

4. (Research + Magic Path dependent)

There is an Earth spell that deal massive damage to Constructs like the Sphynx. Forgot what it is called.

5. (Independent of Nation/Magic Path/Research)

The preach him out of existence method. Note even if you can't preach him out of existence, having a high dominion for you in said province will reduce the Sphynx's hitpoints significantly and increase Moral of your troops, perhaps just enough for you to kill it.

6. (Nation/Magic Path/Research Dependent)

Not exactly a solution, but you can try Curse him, which will make him more likely to accumulate Battle Afflications when swarmed by your troops, which may allow your next attempt to succeed if this one fails.

7. (Nation/Magic Path/Research Dependent)

Magic. There are plenty of offensive magics that can take down a Sphynx relatively easily, especially if your mages are protected from the Sphynx's attacks (e.g. Fire Resistance in your case) Failing that, you can always try to Paralyse the Sphynx if that's an available option. A paralysed target will be easier to hit I believe, especially with Armour Piercing/Negating attacks/spells.

8. (Magic Path/Research Dependent)

Someone suggested in the Teleporting Sphynx thread of using Lamashastas. Certainly viable, but never tried it myself. The summoning of Lamashastas will likely cost your summoners and any other troops around, as the Lamashastas will slay its summoner first, then whatever else is left on the battle field. Can they kill a Sphynx? Having never tried it I can't say for sure, but they sounded quite powerful from other's descriptions.

9. (Research + Magic Path dependent)

Use Bow of Botulf, which if it hits, can Feeblemind the Sphynx, preventing it from casting spells.

10. (Research + Magic Path dependent)

if it is the only unit left, you can also use the Ethereal Bow artifact on a Leader with good precision, tell the leader to "Attack for one turn" how ever many times needed to get sufficiently close, then "Fire". The Bow will kill if you can hit him. Beware of friendly fires!

Just some suggestions that may help you out.

-Gateway103