  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.
