The aim of this article is to determine the dynamical parameters of the asteroid (22) Kalliope with its satellite Linus, based on astrometric observations of this system. We use a model of the motion taking into account the gravitational influence of the primary's axisymmetric non-sphericity. This factor leads to precession of the line of apsides of the satellite, as well as precession of its orbit around the primary's axis of symmetry. The axisymmetric gravitational field of the primary is formed by the time-averaged field of the fast-rotating body of Kalliope. All available observations of the Kalliope-Linus system were divided into groups, each containing observations close in time. For each group, parameters of the fixed Keplerian orbit were determined. Thus, the behaviour of the satellite's orbit axis over about 16-yr time intervals was studied. Contrary to the expected uniform precession of the orbital plane, we discovered abnormal prograde motion of the ascending node in the first seven years, which became retrograde in the next nine years. The amplitude of the change is about 120°. We make an attempt to explain such motion by effects not taken into account in the adopted dynamical model. It is shown that effects not taken into account cannot explain the oscillatory motion of the orbital node found. Using 121 available observations, two formal solutions for all fitted parameters were determined.