Generalized Jacobians of spectral curves and completely integrable systems

Abstract.

Consider an ordinary differential equation which has a Lax pair representation $\dot{A}(x)= [A(x),B(x)]$, where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold \[ \{ A(x): {\rm det}(A(x)-y I)= P(x,y) \} \] 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 pair on the generalized Jacobian is translation-invariant.