So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further assume that the fibers of the map $f:M\to\mathbb{C}^n$ determined by the $f_i$ is an open subset of an abelian variety and the vector fields $X_{f_i}$ are linear. We call such a thing an algebraically completely integrable Hamiltonian system.

Now, I'm told that there's a definition of integrable system in PDEs that acts as some sort of stability condition, though I don't understand it. I've also been told that there's a way to, from an integrable system of PDEs, construct an integrable Hamiltonian system (just drop the condition that fibers be in abelian varieties), and that these two types of objects should be equivalent.

My questions:

1) What's the correct formulation for PDEs to make something like this work out? (I know virtually nothing about PDEs, and would be quite grateful just to be pointed at a good reference, if there's too much I need to read to get a quick, understandable answer)

2) Is there a general method of going from an algebraically completely integrable Hamiltonian system, which is algebro-geometric in nature to working out the PDEs explicitly? Does it help if the symplectic variety is known to be the cotangent bundle of something? How about if the base is unirational? rational?