  We construct a three-parameter family of non-hyperelliptic and bielliptic
plane genus-three curves whose associated Prym variety is two-isogenous to the
Jacobian variety of a general hyperelliptic genus-two curve. Our construction
is based on the existence of special elliptic fibrations with the section on
the associated Kummer surfaces that provide a simple geometric interpretation
for the rational double cover induced by the two-isogeny between the abelian
surfaces.
