KK on paired board

[theblitz]

New at this table so no reads as yet on anyone.

PokerRoom No-Limit Hold'em, $2 BB (6 handed) converter

UTG ($135.80)
MP ($176.60)
CO ($80.50)
Hero ($98.50)
SB ($113.50)
BB ($28.50)

Preflop: Hero is Button with K[img]/images/graemlins/diamond.gif[/img], K[img]/images/graemlins/spade.gif[/img].
<font color="#666666">2 folds</font>, <font color="#CC3333">CO raises to $4</font>, <font color="#CC3333">Hero raises to $11</font>, <font color="#666666">2 folds</font>, CO calls $7.

Flop: ($23.50) J[img]/images/graemlins/spade.gif[/img], 6[img]/images/graemlins/spade.gif[/img], 3[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(2 players)</font>
CO checks, <font color="#CC3333">Hero bets $15</font>, CO calls $15.

Turn: ($53.50) 3[img]/images/graemlins/diamond.gif[/img] <font color="#0000FF">(2 players)</font>
CO checks, Hero ?

At this point I am wondering what could he have that I am ahead of?
QQ is the most likely.
However, I could be badly behind JJ. However, I would have expected him to have bet out or check-raised the flop due to the flush draw.

AA is not an option as I would expect him to have raised again pre-flop and on the flop.

So, check behind or bet out?
What would be my action on the river.

Res of hand to follow.