This is a followup of my previous question Gromov-Witten and integrability. As I have learned from the answer (but guessed before), GW potentials of the point and $P^1$ (with different modifications) are, more or less, the only examples of the GW generation functions with established integrable properties. So what about higher genera curves? Are they really so complicated to establish integrability, at least for stable sector? What is the main problem with them?