  For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings
of elliptic curves E, we give an explicit algebraic description of all
birationally non-equivalent genus 2 curves whose Jacobians are degree 2
isogeneous to the Prym varieties associated to such coverings. Our description
is based on previous studies of Prym varieties with polarization (1,2) in
connection with separation of variables in a series of classical and new
algebraic integrable systems linearized on such varieties.
  We also consider some special cases of the covering C -> E, in particular,
when the corresponding Prym varieties contain pairs of elliptic curves and the
Jacobian of C is isogeneous (but not isomorphic) to the product of 3 different
elliptic curves.
  Our description is accompanied with explicit numerical examples.
