Given that he's solid:
He must have KK or AK about 80% of the time. Maybe the other 10% he has JJ-QQ, and 10% air.

Once you call his flop bet, he must think that you will not call another bet unless you beat KK.

I don't think he has AA or set here because he would bet again, hoping you will call with an overpair. On the otherhand, I doubt he folds KK (or even the rare QQ). I would also think he might bet air again, since that's the only way he can win.

Putting the rest in folds AK and anything else you beat. You are giving him a chance to draw to 6 outs... but most of the time you are crushed.