Generalized Jacobians of spectral curves and completely integrable systems

Consider an ordinary differential equation which has a Lax pair representation A'(x)= [A(x),B(x)], where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only onA(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold {A(x): det(A(x)-y I)= P(x,y)} of this Lax pair is an affine part of a non-compact commutative algebraic group---the generalized Jacobian of the spectral curve {(x,y): P(x,y)=0} with its points at "infinity" identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pairon the generalized Jacobian is translation--invariant. We provide two examples in which the above result applies.