  It is known that the Jacobian of an algebraic curve which is a 2-fold
covering of a hyperelliptic curve ramified at two points contains a
hyperelliptic Prym variety. Its explicit algebraic description is applied to
some of the integrable Henon-Heiles systems with a non-polynomial potential.
Namely, we identify the generic complex invariant manifolds of the systems as a
hyperelliptic Prym subvariety of the Jacobian of the spectral curve of the
corresponding Lax representation.
  The exact discretization of the system is described as a translation on the
Prym variety.
