3/6 NL KK, What is optimal?

Here is a hand that may appear to be fairly standard, but I am not 100% sure what the most profitable play is long-term.

I do not have hand history with me, no major reads on villain, with about 80 hands against him, he is approx 28/8.

Effective stacks $700...Villain is in SB and it folds to me on the button with KK. Standard raise to $18, and villain reraises to $42. I just call, not wanting to give away too much info. Flop comes 939r. Villain leads for $60, I pop it to $160 (should raise more?) Villain thinks for a while and just calls. Turn comes a 2, villain checks and I bet 1/2 pot and villain shoves.

I guess my question is more whether I should have just shoved flop or turn?

Am I really up against Aces with this guy? I was thinking he would be on JJ/QQ, but always the distinct possibility I am up against AA here.

Thoughts on all streets appreciated.

Your image is a big part of this. If you've been raising a lot and it's looking like a typical blind steal, SB will give you a lot less credit for a real hand.

The flop is pretty useless and your reraise could mean nothing since it's HU (this goes back to your image).

TT JJ QQ are very likely here. Hell, even 88 or 77 could be in play here.

More times then not I don't fear AA here and call the shove.

I hadn't been getting out of line too much, and showing down winners when getting aggressive. I was playing approx 20/10 at this point. Perhaps he views it as a steal. I don't think I can put him on any sort of a 9 given the RR unless he's really coming after my "steal" attempt.

Would you think I should just shove on the turn rather than let him c/r me and I take the initiative?

You have played this with a standard AK/AQ/KQ line imo. Raise then call preflop, then raise the flop. You then bet the turn to see if you can take the pot. Looks standard. He has no way of knowing you have a monster.

I give A9 no credit here, but 89, 9T, or 79 sooted are possible.

At lower stakes I fold this most times, but having looked at many 600NL hh's, I don't see myself getting away from this, nor playing it any differently.

it's fine