Generalized Jacobians of spectral curves and completely integrable systems

M P = {A(x) ∈ M : det(A(x)− yIr) = P (x, y)}. The system (1) has an obvious symmetry group G = IPGLr(I C; J) which is the subgroup of the projective group IPGLr(I C) formed by matrices which commute with J . The group G acts on M by conjugation, the action is Poisson, and the reduced Hamiltonian system is completely integrable too. As the symmetry group G acts freely and properly on the general isospectral manifold M P , then M J P can be considered as the total space of a holomorphic principal fibre bundle ξ with base M P /G, structural group G, and natural projection map M P φ → M P /G.