  A correspondence between 1) rank 2 completely integrable systems of Jacobians
of algebraic curves and 2) (holomorphically) symplectic surfaces was
established in a previous paper by the first author. A more general abelian
variety that occurs as a Liouville torus of integrable systems is a prym
variety associated to a triple (S,W,V) consisting of a curve S, a finite group
W of automorphisms of S and an integral representation V. Often W is a Weyl
group of a reductive group and V is the root lattice. We establish an analogous
correspondence between: i) Rank 2 integrable systems whose Liouville tori are
generalized prym varieties Prym(S_u,W,V) of a family of curves S_u, u in U. ii)
Varieties X of dimension 1+dim(V) with a W-action and an invariant V-valued
2-form. If V is one dimensional X is a symplectic surface. We obtain a rigidity
result: When the dimension of V is at least 2, under mild additional
assumptions, all the quotient curves $S_u/W$ are isomorphic to a fixed curve C.
This rigidity result imposes considerable constraints on the variety X: X
admits a W-invariant fibration to C and the generic fiber has an affine
structure modeled after V. Examples discussed include: Hitchin systems, reduced
finite dimensional coadjoint orbits of loop algebras, and principal bundles
over elliptic K3 surfaces.
