To study the way in which the principal periodic orbits in a Galactic potential determine orbital structure, horizontal and vertical surfaces of section, i.e. (dR/dt, R) and (dz/dt, z), are being used to explore the potential of Allen & Santillán (1991) and to investigate possible vertical structure in the Galactic halo. The chaotic “scattering” process due to the nearly spherical mass distribution close to the Galactic center in conjunction with the confinement of the chaotic orbits produces a vertical segregation of both chaotic and non-chaotic orbits in the halo. Certain z max , z min are preferred by the chaotic orbits over others as a result of the conservation of the total orbital energy and of the interaction and confinement of the chaotic orbits by the principal families of periodic orbits (Figure 1). Some of these periodic orbits have been identified. Correlations between the structure found in the observed W distribution and that of the numerically determined z max , z min histograms are shown for our sample of 280 halo stars (Schuster et al. 1993). W is the star's velocity perpendicular to the Galactic plane and z max , z min the maximum distances above or below the Galactic plane, respectively, reached by the star in the course of its orbit. This vertical structure may explain certain puzzling observations of the galactic halo, such as conflicting c/a values for the shape of the halo, and unusual velocity dispersions and/or distributions near the Galactic poles. These results are in good agreement with Hartwick's (1987) two component model for the halo.