In real games, it seems an NLH heads up bot is now at the point of (or close to) being able to beat the best human players. http://www.cs.cmu.edu/news/cmu-ai-tough-poker-player
For heads up limit, the Nash equilibrium is close to known http://arstechnica.com/science/2015/01/computers-used-to-solve-two-person-limit-texas-holdem/
Logically, if each situation, defined by having the exact same cards on the table and the exact same preceding action, has only one bet sizing, NLH is no larger than limit holdem. For example, if there's a bet and a call preflop and the flop comes 842 all hearts, the continuation bet (assuming there is one) could always be 3/4 pot. I assume no mixing between bet sizes would mean that bet sizes are independent of the hole cards, but I could be wrong.
In addition, if there's no mixing between different actions, no 80% bet / 20% check, but always 100% one or the other, the game becomes significantly smaller.
Of course, a strategy without mixing is significantly worse than the true Nash equilibrium, but it would still be interesting to see. Is such a strategy known? Is there any publicly available software that calculates such a strategy?