  The Bertrand-Darboux integrability condition for a certain class of perturbed
harmonic oscillators is studied from the viewpoint of the
Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the
BG-normalization on computer algebra, it is shown that if the perturbed
harmonic oscillators with a homogeneous-{\it cubic} polynomial potential and
with a homogeneous-{\it quartic} polynomial potentials admit the same
BG-normalization up to degree-4 then both oscillators satisfy the
Bertrand-Darboux integrability condition.
