  A pure two-body problem has seven integrals including the Kepler energy, the
Laplace vector, and the angular momentum vector. However, only five of them are
independent. When the five independent integrals are preserved, the two other
dependent integrals are naturally preserved from a theoretical viewpoint; but
they may not be either from a numerical computational viewpoint. Because of
this, we use seven scale factors to adjust the integrated positions and
velocities so that the adjusted solutions strictly satisfy the seven
constraints. Noticing the existence of the two dependent integrals, we adopt
the Newton iterative method combined with the singular value decomposition to
calculate these factors. This correction scheme can be applied to perturbed
two-body and N-body problems in the solar system. In this case, the seven
quantities of each planet slowly vary with time. More accurate values can be
given to the seven slowly-varying quantities by integrating the integral
invariant relations of these quantities and the equations of motion. They
should be satisfied with the adjusted solutions. Numerical tests show that the
new method can significantly reduce the rapid growth of numerical errors of all
orbital elements.
