On the existence of a third integral of motion

Abstract : Different methods for finding integrals of mo tion are discussed. The third integral is ex plicitly calculated by means of von Zeipel's method. Different definitions of integrable systems are given and a distinction between use ful and nonuseful integrals is made. Poincare's nonexistence theorem is mentioned and its many exceptions are pointed out. Then a distinction of separable and nonseparable systems is made. The problem of the convergence of the third in tegral is discussed. Numerical results provide ample evidence that the third integral exists in quite general potential fields, ain Schmidt's model of the Galazy, in slightly elliptical clusters or galaxies, in the field of the oblate earth, in resonance cases where the unperturbed frequencies have a rational ratio, and in po tential fields that do not have a plane of sym metry. In the case of non-axially symmetric sys tems two new integrals are introduced. However, no integral corresponds to the angular momentum. The third integral can be applied to the bound aries of the orbits and their hodographs, to the study of periodic orbits, to the dispersion of a group of stars, to the three-axial velocity el lipsoid, to the construction of models of the Galaxy, etc. Finally a practical method for searching for new integrals is indicated. (Author)