In the present work, we define a new anomaly, $\Psi$,
termed semifocal anomaly. It is determined by the mean between the true
anomaly, $f$, and the antifocal anomaly,
$f^{\prime}$; Fukushima defined
$f^{\prime}$ as the angle between the periapsis
and the secondary around the empty focus. In this first part of the
paper, we take an approach to the study of the semifocal anomaly in the
hyperbolic motion and in the limit case correspoding to the parabolic
movement. From here we find a relation beetween the semifocal anomaly
and the true anomaly that holds independently of the movement type. We
focus on the study of the two-body problem when this new anomaly is used
as the temporal variable.\\ In the second
part, we show the use of this anomaly —combined with numerical
integration methods— to improve integration errors in one revolution.
Finally, we analyze the errors committed in the integration process
—depending on several values of the eccentricity— for the elliptic,
parabolic and hyperbolic cases in the apsidal region.
Analytical solution of the two-body problem against the cosmic vacuum background
