Coffey, ShannonDeprit, AndréDeprit, EtienneHealy, LiamImages in article have degraded quality in this reproduction. High quality cover image included.For an integrable dynamical system with one degree of freedom, "painting" the integral over the phase space proves to be very effective for uncovering the global flow down to minute details. Applied to the main problem in artificial satellite theory, for instance, the technique reveals an intricate configuration of equilibria and bifurcations when the polar component of the angular momentum approaches zero.en-USphase spaceHamiltonian systemintegrableone degree of freedomvisualizationsatellite orbitArticle