Generalized Jacobians of spectral curves and completely integrable systems

Consider an ordinary differential equation which has a Lax pair representation , where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold of this Lax pair is an affine part of a non-compact commutative algebraic group – the generalized Jacobian of the spectral curve with its points at “infinity” identified. Moreover, for suitable B(x), the Hamiltonian vector field defined by the Lax pair on the generalized Jacobian is translation-invariant. Received April 29, 1997; in final form September 22, 1997