  Let F be a real quadratic field, and let R be an order in F. Suppose given a
polarized abelian surface (A,\lambda) defined over a number field k with a
symmetric action of R defined over k. This paper considers varying A within the
k-isogeny class of A to reduce the degree of \lambda and the conductor of R. It
is proved, in particular, that there is a k-isogenous principally polarized
abelian surface with an action of the full ring of integers of F when F has
class number 1 and the degree of \lambda and the conductor of R are odd and
coprime.
