  We prove that for any autonomous 4-dimensional integral system of Painlev\'e
type, the Jacobian of the generic spectral curve has a unique polarization, and
thus by Torelli's theorem cannot be isomorphic as an unpolarized abelian
surface to any other Jacobian. This enables us to identify the spectral curve
and any irreducible genus two component of the boundary of an affine patch of
the Liouville torus.
